Tarzia has publicly conceded he first heard of “allegations concerning the conduct” of the Waite MP “near the end of December”, while Premier Steven Marshall revealed this week he “was made aware of this issue when it occurred”, on December 13.

“All I would say is there have got to be consequences for this type of behaviour… now [Duluk’s] suffered a very serious penalty and there’s a further investigation underway,” the Premier told media on Monday.

Duluk has since stood down from two parliamentary committees and will seek counselling over his use of alcohol, after further claims were aired by Greens MLC Tammy Franks, as InDaily revealed yesterday.

But Tarzia, who on Tuesday told InDaily “it will be a priority to appoint a third party” to oversee an inquiry into the allegations, is yet to announce any such appointment, or the scope or nature of the investigation.

The Opposition today accused the Government of dragging its feet, with frontbencher Tom Koutsantonis saying: “The Premier said an investigation was underway – if it’s underway, who’s conducting it?”

“What are the terms of reference? Do they have the power to call and summons witnesses? And once the investigation is complete, who is it referred to?” he said.

“The Premier’s known about it since mid-December… why has it taken so long?”

Tarzia did not return calls today, instead responding to several questions by SMS: “Nothing further to add at this stage, will be in touch when we do.”

Equal Opportunity Commissioner Niki Vincent, whose office had discussed the appointment of an independent investigator with the Clerk of the House, said she had “no idea of what’s going on” with the inquiry.

She said she had not been told anything further, adding: “I’d say I would have been if it was going to be that I was asked [to conduct it].”

Treasurer Rob Lucas said any funding for an inquiry would come from the House of Assembly’s own independent allocation.

Duluk said yesterday he had “assured the Premier and the Speaker that I intend to fully co-operate with any investigation thought necessary and appropriate”, publicly reiterating “my apologies to those whom I offended that night”.

Labor has called for Marshall to demand Duluk’s resignation from parliament and move to expel him from the Liberal Party.

But Franks said if the MP were to shift to the crossbench – reducing the Government’s numbers on the floor of the House to 23 seats – it should not be used as a political tool by the Opposition in parliament.

“If the parliament’s serious about this, and the Liberal Party takes action as a result of this… all sides of politics need to remove the numbers from the equation,” she said today.

“Labor needs to make it clear they’ll not accept his vote on a substantive or procedural motion before the parliament… they need to guarantee they won’t benefit politically from that, should it occur.”

Koutsantonis said he had no comment about whether Labor would accept Duluk’s vote should Marshall heed the Opposition’s calls to remove him, saying he would need to consult colleagues.

“Right now, the only person accepting his vote and his support is the Premier,” he said.