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Conservative Jim Campbell was silenced by Cllr McVey at the city council's policy and sustainability meeting and then rebuked when he posted a message saying: "Free speech now being curtailed".
Now he has likened it to the scenario which made Jackie Weaver an internet hit during lockdown after she was called in to chair a fraught parish council meeting and ended up muting and then virtually ejecting bickering members, provoking protests of "You have no authority here".
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Cllr Campbell said Cllr McVey's muting of him was "an extreme response". He said: "Our standing orders say nothing about muting a member who wishes to speak. Rather like Jackie Weaver, I suspect there is no basis on which a convener has the authority to do this."
The muting came after Cllr Campbell tried to ask officers if they felt the decision of the previous full council meeting not to debate a damning report on the council's secure accommodation met an audit report recommendation about the council "better demonstrating how it responds to feedback and scrutiny findings."
He said: "If Adam had said 'I think that question isn't appropriate I think you need to rephrase it', I would have tried to. I'm quite confident officers could have handled it – they'd probably have said 'It's not really our place to answer that' and that would have been the end of it.
"Standing orders have been updated to reflect meetings moving online but they have not said anything specific about the technology and how it can be used. They are silent around that – in the same way that Jackie Weaver has pointed out people don't really know what the powers are."
Cllr Campbell said some colleagues thought he should formally challenge Cllr McVey's use of the mute button, but he said it was "not my style".
Six separate reports published last week ruled that Jackie Weaver's actions were "without authority".
Cllr Campbell said: "I think that would probably be the same if it were challenged up here."
Cllr McVey hit back at Cllr Campbell’s comments. He said: "At every level the rule-breaking behaviour from the Tories has got worse. If Cllr Campbell is keen to address this behaviour, he should start by setting an example and following the rules himself, not moaning about being made to adhere to basic standards of conduct.”
A council spokesperson said: “Standing Orders state that the convener has the power to decide on all matters of protocol, decorum, order, competency and relevancy. If the convener decides that a question by a member is not relevant to the report being discussed or is inappropriate then they can move on to another question. The convener can mute members to keep the business of the committee progressing and ensure committee's conduct is within the rules of standing orders and code of conduct."