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The controversial £120 million scheme looks set to be ratified with a timetable that would not see the project completed until 2027.
But Edinburgh's depute council leader Cammy Day said he was "not convinced" the plans were the right thing for the Capital and voiced concern about more cars coming into the city.
The project – funded through the Edinburgh City Region Deal – is aimed at tackling peak-hour bottlenecks by using the flyover to separate local traffic from through traffic.
If Friday’s meeting of the City Region Deal joint committee, made up of council leaders from Edinburgh, the Lothians, Borders and Fife, approves the scheme, a public inquiry is expected in early 2022 with a ministerial go-ahead in late 2022.
This would lead to the tendering procedure starting in spring/summer 2024, the contract being awarded in spring 2025, construction work getting under way in summer 2025 and the flyover opening in late 2027.
A report to the committee says in addition to easing congestion, the scheme will also "provide high-quality active travel facilities to address severance challenges and deliver a step change in accessibility for walking and cycling" as well as improving road safety and improving journey times for all road users, including buses.
But Councillor Day, who is leader of Edinburgh's Labour group, said: "My concern is that this could increase the traffic coming into the city at a time when we're trying a whole range of measures to reduce vehicles coming in and out of the city and promote active travel with our commitment to 2030 net zero. This, for me, doesn't quite fit.
"I'm not convinced it is the right thing for the city. I absolutely understand the need to respond to the congestion around Sheriffhall and we need to do everything we can to promote public transport and active travel – and that should be the solution, not to build flyover into the city."
The report said traffic modelling suggested a 6 per cent increase in traffic on Old Dalkeith Road coming into Edinburgh, but that would be balanced by a decrease in traffic adjacent routes.
Meanwhile, Claire Miller, transport spokesperson for the Green group on the city council, questioned why the project was going ahead despite the recent deal between the SNP and the Greens at Holyrood.
She said: “The deal struck between the Greens and the government is clear – we won’t see more unnecessary road building in Scotland. So I am very surprised to see council leaders seem to be ignoring the government and carrying on regardless.
"I’m calling for [Edinburgh council leader] Adam McVey to uphold the deal his party has agreed with the Greens, put a stop to this expensive and outdated project, and invest the cash in pavements, cycle ways and public transport.”
Cllr McVey said a key element of project was for all partners to work together to minimise traffic travelling across boundaries in the region and support public transport and active travel alternatives.
“We need a bypass that works for the region and Sheriffhall is an important part of that,” he said.
"Over recent years we’ve made strong representations on active travel and public transport and have spoken in detail with Transport Scotland, public transport operators, including Lothian Buses, active travel groups and community representatives on these plans.
"Many issues have been addressed and it’s clear the project will improve active travel, relieve congestion and encourage traffic onto the bypass and out of residential areas of Edinburgh.”
Transport Scotland said the scheme included nearly two miles of segregated shared surface for pedestrians and cyclists, with five dedicated subways for pedestrians and cyclists under the new roundabout and A720, linking up to the A6106 North and South and the A7 North and South.
“The scheme will also encourage uptake of bus through improved links to adjacent Park and Ride facilities,” the transport body said.
"In addition, the main A720 structures have been designed to accommodate any future plans for bus priority or tram provision within the scheme extents."
And it pointed to the passage in the policy agreement with the Greens, which said work on trunk roads projects “under construction, design, development or procurement” would continue, subject to normal statutory assessment and business case processes.