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The £120 million scheme to ease traffic flow at one of the most congested junctions on the Edinburgh City Bypass was awarded funding under the Ednburgh City Region Deal in 2018.
But preparatory work on the project was halted in February 2020 after the Greens called for it to be reviewed as part of their Holyrood budget deal with the SNP.
Now Mr Briggs says he fears the Greens will press for the plans to be abandoned during their talks on closer co-operation with the Scottish Government which are taking place over the summer.
He said there had been no indication from the government of how the review was progressing or what was going to happen to the project.
“It increasingly looks like it's going to stall and we'll just be left with a nightmare roundabout.
“The traffic is increasing – I think even during the lockdown there haven't been many times when there haven't been back-ups at this junction. It’s a major issue for the whole region which isn’t being addressed by the government.
"All the communications I’ve had from ministers are just stalling – it feels like it's become ‘Sheriffstall’ under the SNP.”
And he asked: “Has this become part of the SNP-Green coalition deal they’re trying to put together?
"They used it during a budget negotiation to get this review and Greens in Edinburgh and Midlothian seem to have been dead against any moves to upgrade it.
“All the discussions between the Greens and the SNP are behind the scenes and secret, but this project has been delayed by their shenanigans for too long anyway."
The flyover is intended to separate local traffic from through traffic on the bypass and allow bypass traffic to flow freely, improving journey times for all road users.
But Greens branded it a "spaghetti junction", claiming it would just generate more traffic.
The city council has also said it wants major changes to the design to allow for bus lanes and in the longer term an extension of the Capital's tram route.
Mr Briggs said: “I'm keen we see safer cycle routes as part of the upgrade as well, but the fact nothing is happening at all isn’t much use to anyone.
“It is after a key arterial link into the city and it’s already over capacity on that key stretch of road.”
The Greens declined to say whether Sheriffhall would feature in the talks, which are focusing first on broad areas for co-operation, though these include infrastructure spending.
But a party spokesman said: "Scotland’s response to the climate emergency will be a key component of our co-operation discussions with the Scottish Government.”
A Transport Scotland spokesman said: “We are committed to undertaking improvements at Sheriffhall as part of the Edinburgh and South East Scotland City Region Deal and continue to progress the scheme through the statutory procedures.
“The review into public transport and active travel in the Sheriffhall scheme has concluded and confirmed that the scheme provides significant benefits for both modes of transport. Overall we are satisfied that the scheme removes the barrier that the current junction forms across multiple modes and to journeys of national, regional and local importance.
“As the scheme has received a number of objections we are continuing to consider and respond to those as is our statutory responsibility.”