Kuusik resigned after one day in the post, at the beginning of May. His resignation came in the wake of allegations of domestic violence, now the subject of a Police and Border Guard Board (PPA) investigation, and pressure including President Kersti Kaljulaid's refusal to be present at his swearing-in at a Riigikogu ceremony.

However, a substitute candidate has yet been nominated to the role.

''People are one by one not wanting to be asked questions which might reveal things about them; they don't want to be caught 'naked','' Helme said at Thursday morning's regular government press conference, pointing to the keen media interest in the likely replacement candidate.

He added that public mistakes made 10-15 years ago, for example traffic offenses and the like, will be reported as if they are recent events and thus be used in potentially eliminating candidates.

Prior to the domestic violence allegations, it emerged that Marti Kuusik had on the morning following the March 3 general election been stopped by the PPA in the town of Rakvere. He had been speeding at over 20 kmh in excess of the speed limit, and was found to have trace alcohol in his system, following a breathalyzer test.

''It is true that several people have turned down the offer, noting that they have no desire to be a target for being spat on,'' Helme, who was standing in for an absent Jüri Ratas (Centre) went on.

He noted that there were nonetheless candidates on the table, but that noone would be named until they had been presented to the prime minister.