The Gorgie fans had tried to tether their own expectations by reminding themselves of that.
But having seen the Serie A side drop points to both RFS and Istanbul Basaksehir in their previous two matches, to trail the Scottish side in Group A, there was always hope.
A win would have opened the door to the home side as they continue on their quest to progress to the next stage of the competition and would have allowed them to travel to Florence next week in a positive frame of mind.
But now they head to Tuscany playing catch up and they do so with an ever-more depleted squad, after young Lewis Neilson became the latest Hearts player to see red in a big game.
Just as Cammy Devlin’s sending off against Rangers did not cost them a game that had already scampered away from them, the numerical disadvantage caused by the former Dundee United defender’s dismissal against Fiorentina was not Hearts’ downfall.
On both occasions they were already 2-0 down before the referee stepped in. and, on both occasions they contributed to their own downfall.
If poor individual performance were to blame against Giovanni van Bronkhorst’s side, against the Italians, tactics and personal choices were certainly up for debate and will have contributed to the boos that harshly rang around the stadium at half-time.
Up against a technically-superior side, even Pasquale Bruno, the former Hearts and Fiorentina midfielder had warned that the capital side would not be able to beat their guests in a pure footballing contest. But he knew the impact the Scottish style of play can have on rivals.
The hope was that Hearts could dictate a high tempo and get in about Vincenzo Italiano’s men and make life uncomfortable for them. To the thrill of the sell-out Tynecastle crowd, that was they way they started the match.
But, as time moved on they were remiss when it came to pressing as a unit, always leaving an escape route for the Italians who, as a consequence, rarely looked ruffled.
Given the way they had provided the dig and energy against Rangers at the weekend, it was therefore surprising that neither Devlin nor Stephen Humphrys made the starting line up.
Two players with a gallus streak, who rarely stand on ceremony, too respectful of big name opponents, without them the travelling side enjoyed too much space and time in the middle of the park where they dictated the tempo and the pulled the strings.
Hearts manager Robbie Neilson, recognising the respective investment in the teams and the wealth of talent at his counterpart’s disposal had not tried to match them in the purest terms, foresakiing any attempt to go for free-flowing football through the middle and instead try to ping long diagonals aimed at unleashing rapid wingers Josh Ginnelly and Barrie McKay down the flanks.
It was a tactic that did not trouble the away side unduly.
If the match against RFS had given Hearts cause to celebrate, an early goal for Fiorentina knocked the self-belief of a side who could have used the likes of Craig Halkett, Liam Boyce and Beni Baningime against such lofty opponents.
There were only five minutes on the clock when Aleksa Terzic found room down the right and played a cross in for Rolando Mandragora to head back across Craig Gordon and into the net.
Hearts were struggling to marshall runners from midfield, while the guests’ movement in the box bought them space to punish Hearts.
Hearts’ longball tactic almost reaped dividends in the ninth minute when a ball in from McKay in the corner found Michael Smith but his header was too close to Pierluigi Gollini.
Hearts were showing too much respect, backing off Fiorentina and sitting too deep. Those spaces are the ones normal patrolled by the effervescent Devlin and his dynamic presence was a miss as the way side dominated the possession
But Hearts showed that their rivals could be rattled, if they had a go, but it took until late in the first half to really test that, forcing Gollini to clear into section N before Ginnelly finally got the break and with a clear sight of goal, had his shot blocked.
But at least there was a shot. Until then the Fiorentina backline had managed to cut out the attempted passes through.
But Ginnelly, Jorge Grant, Kio and McKay had all linked up. And Grant was close to sending Lawrence Shankland clear for a shot but, once again, a Fiorentina leg foiled the home outfit.
It was for nothing, though, as the away sides made it 2-0 before the interval.
Neislon didn’t track his runner and by the time the cross came back in and Christian Kouame got over it to athletically bury his shot past Gordon, he had about five Hearts players around him. But the service and the movement was too good for them.
At 2-0 there was probably no way back and boos serenaded them off the pitch. But when they reemerged for the second half, the early sending off, as Neilson stepped in to deny Jovic a clear goalscoring opportunity in the 48th minute, their fate was sealed.
If Fiorentina had enjoyed too much space before, they were able to revel in it and had they not been a bit too casual they might have made things more embarrassing.
As it was, Humphrys and Devlin were both thrown on and they did make a bit of a difference, helping to negate the one-man disadvantage, if nothing else.
But they couldn’t stop Fiorentina sealing the win that allowed them to leapfrog Hearts and move closer to the Turks who drew with RFS in the group’s other match as Jovic made it 3-0 after Kouame fired in the cross.