The corporate regulator has agreed to investigate further the business affairs of the Liberal MP Stuart Robert, at the request of Labor.

Robert, the assistant treasurer – who recently blamed “connectivity issues” for charging taxpayers nearly $38,000 for his home internet usage since 2016, before agreeing to repay it – will face deeper scrutiny of his role with Cryo Australia.

Officials from the Australian Securities and Investments Commission (Asic) have agreed to consider if Robert breached the Corporations Act by failing to notify Asic in time of his recent resignation from Cryo Australia.

Earlier this month, Asic records showed Robert was still a director of Cryo Australia (a cryotherapy company that uses freezing temperatures for supposed health benefits) even though it had been a month since he had told parliament that he had quit the board after Scott Morrison invited him back into the Coalition ministry.

When the the Weekend West inquired this month why Asic records showed Robert was still a director of Cryo, the company reportedly lodged documents with Asic within hours showing Robert had quit as a director. A spokesman for Robert then told the paper that Robert had resigned from the board “with effect” from 31 August.

Asic officials say they will investigate whether Robert breached the Corporations Act by failing to notify them of his resignation from Cryo within 28 days.

Asic official Warren Day said it was not uncommon for directors to publicly announce their resignation and for the lodgement to take time to reach Asic’s register.

In Senate estimates hearings on Wednesday evening, Asic officials were also asked about the circumstances surrounding Robert’s involvement with Robert International, another one of his business affairs.

Fairfax Media reported in September last year that Robert’s father, Alan Robert, was unaware that he had been made a director of a private investment company that held shares in his son’s IT service business which had won millions of dollars worth of government contracts. It was reported that Alan Robert said the company, Robert International, was actually run by his son for the six-year period that he and his wife, Dorothy, were listed as the company’s sole directors without their knowledge.

At the time, Robert accused Fairfax Media of publishing “a load of rubbish” and attacked Labor for referring the matter to the corporate regulator.

He claimed Fairfax had lied about speaking to his father but later accused Fairfax of asking his father “opaque” questions. He also said his father was caring for his mother, who was recovering from a heart attack.

Asic made initial inquiries at the time but decided not to pursue the matter, saying the allegations related to forms lodged with Asic in 2010 so there would be statutory and evidentiary limitations on Asic’s ability to pursue it. It also said Alan’s and Dorothy’s age and health suggested there would be further limitations on any investigation, and the matter involved director appointments of a small proprietary company so it would be unlikely to affect consumers or investors in the broader economy.

On Wednesday evening, the Labor senator Chris Ketter asked Asic officials why they had not pursued the matter beyond its initial stages.

Day said Asic had reviewed a number of documents relating to Robert International, including the listing of directors’ names, and it considered media reports about the age of Robert’s parents and believed it did not need to pursue the matter further.

“Given their age and their health, that would impact on their memory and other factors in relation to what they knew they did sign or did lodge at those times,” Day said.

Ketter asked if Asic had tried to independently confirm how healthy Robert’s parents were, given the assumption that their ill health would prevent them from assisting an investigation of the matter.

Day said: “From the documents we saw, that confirmed their age, yes.”

Ketter asked: “What about their health?”

Day replied: “No.”

Ketter said: “So you just accepted that?”

Day replied: “Yes.”

The shadow assistant treasurer, Andrew Leigh, said the allegations discussed in Senate estimates were troubling.

“The public deserve to know that everything has been done to ensure that Mr Robert has done no wrong, particularly since the allegations surrounding his involvement as a director of companies would fall under his responsibilities as a government minister,” he said.

“We welcome Asic’s confirmation that they will make a new series of inquiries into Mr Robert’s directorships.”