Its official: the Cook Islands Parliament will not meet until September. Which means a government which has lost the confidence of the House and therefore has no mandate to govern will be able to collect its salaries and perks and limp on by the simple expedient of proroguing democracy.
This is possible because the Cooks still effectively runs on the Royal Prerogative. Legal authority flows down from the monarch rather than up from the people, and Parliament is still seen in its C16th way as an adjunct to the government's will rather than as the true source of legitimacy. Its a strong argument against such constitutional arrangements.
Fortunately, it can't happen here. The NZ Parliament is not at the beck and call of the Prime Minister, but sets its own timetable by motion every year. While a Prime Minister can advise the Governor-General to prorogue Parliament and delay its sitting, by convention that now requires the confidence of the House (as does calling elections). So a PM who loses confidence over the summer break cannot escape their democratic doom.