The potential referral of Peter Dutton to the high court is in doubt after Andrew Wilkie insisted MPs facing similar constitutional questions should be referred together and Kerryn Phelps reserved her position.

Although Wilkie and other crossbench MPs deflected government threats to send Phelps to the high court, the insistence that a group of three Labor and Coalition MPs be referred lessens the chances that the home affairs minister will be sent to test if he has a financial conflict of interest.

While the numbers aren’t there yet for a referral, the crossbench has thrown down the gauntlet to the government before the last week of parliament on issues including live sheep exports, medical transfers from Nauru and blocking public funds from indemnifying coal power plants.

Labor has yet to reach a position on the immediate transfer of children and their families from Nauru but even with Labor support the Phelps’ bill lacks the crucial two Coalition votes needed to suspend standing orders and pass legislation over government objections.

After losing its majority with the election of Phelps, the government is vulnerable to motions requiring a simple majority but can block legislation on those issues unless an absolute majority of 76 is achieved.

On Thursday the Senate agreed to debate Labor’s bill to repeal religious exemptions that allow discrimination against LGBT students and vote on it on Monday, one bill that could challenge the government in the lower house next week if Coalition MPs are prepared to cross the floor.

But momentum for a national integrity commission stalled when the Senate voted to refer a bill for an anti-corruption body to a Senate committee inquiry to report by 5 April.

Senator Jacinta Collins told the Senate although Labor’s position was to establish an anti-corruption body in its first 12 months of government “it should not be rushed”.

The Greens democracy spokeswoman, Larissa Waters, said the move meant the opposition had kicked the issue “into the long grass”, with the bill now unlikely to be debated and passed before the Coalition dissolves parliament shortly after the 2 April budget.

Earlier, Phelps convened a press conference of crossbench MPs to announce she will introduce a bill on Monday to remove all the children and their families from Nauru.

She told reporters that Australians want to see the “humanitarian crisis” resolved in a bipartisan way.

“It is really about getting the right kind of medical and psychological care for people who are suffering,” she said. “I can’t see a reasonable argument against this.”

On the issue of referring Dutton to the high court over his family trust’s ownership of childcare centres in receipt of federal subsidies, Wilkie said: “Everyone who is in doubt should be referred … and they should go as a job lot.”

Wilkie said the MPs with “financial question marks over them” – for alleged indirect pecuniary interests in agreements with the commonwealth – were “at least” Dutton, the Liberal Chris Crewther, Labor’s Tony Zappia and the National David Gillespie.

“As much as a lot of people would like to see Peter Dutton go to the high court, we shouldn’t be blinded by emotion, we should be guided by high principle.”

The solicitor general has advised that Dutton is “not incapable” of sitting but there is “some risk” the high court would disagree.

The Centre Alliance MP Rebekha Sharkie said she would support referrals where the solicitor general’s advice “raises a question with respect to section 44”, a standard that could require further advice on Crewther, Zappia and Gillespie before a group referral.

“So far, to my knowledge, we have Peter Dutton in that category,” she told the Australian.

On Wednesday the government leader in the house, Christopher Pyne, threatened to retaliate against Phelps and two Labor MPs if parliament decides to refer the home affairs minister.

Wilkie and the Greens MP Adam Bandt both said they did not believe that MPs who are or were GPs – including Phelps and Labor’s Mike Freelander – have a serious question to answer, as Medicare rebates are collected by doctors on behalf of patients.

Phelps said she is “very seriously considering” all issues before her including section 44 referrals to the high court.

She said each case had to be taken on its merit and “conflating one issue” with another “is a political threat and the Australian people can see right through that”.

Bandt intends to introduce a bill, seconded by Wilkie, to prevent the government from using public funds to indemnify coal power plants, defining it as an “improper purpose”. Phelps said she agreed federal funds should not go to coal power plants.

Contributor

Paul Karp

The GuardianTramp

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