Kilman only came to power earlier this month after bringing a successful vote of no confidence against Jo Natuman's government.
But Natuman's group has contested the legality of that and is seeking to overturn it on the grounds that most of Kilman's supporters are facing charges of bribery.
He also claims the dumping of his government has interfered with the recovery from Cyclone Pam.
The Natuman group's motion for a vote had initially been rejected by the Speaker, Marcellino Pipite, but they successfuly sought a Supreme Court ruling that their constitutional rights had been denied.
Pipite then appealed that ruling.