PIL argues there has already been too long a delay and only a fully independent inquiry can deliver justice.
On Thursday the IPCC revealed it had rejected those findings and ordered a new and fully independent inquiry.
The company needs a fully independent inquiry into what happened, a critical assessment of its current businesses and strategy and an objective review of governance, among other things.
Johnson's father wanted "a fully independent inquiry, carried out by a figure wholly unconnected with the party, which will examine allegations and publish findings.
A company truly committed to restoring the trust of customers and shareholders would have addressed accusations of boardroom cronyism by commissioning a fully independent inquiry.
But Mr Stafford Smith told the BBC's Breakfast on Saturday: "We had a promise from the Prime Minister that there would be a fully independent inquiry … Unfortunately that's not happened yet.
In July 2010 when he announced the "fully independent" inquiry, David Cameron had said that to ignore the claims of wrongdoing would risk secret operatives' reputation "being tarnished".
It suggested a fully-fledged inquiry may follow.
The suspect reported the suicide to the police and fully assisted inquiries.
On his website, Portsmouth South MP Mike Hancock said he went voluntarily to a police station on Tuesday and "co-operated fully" with inquiries.
