There are only eight active coronavirus cases in Tasmania, Public Health Services director Mark Veitch said on Tuesday morning.
Dr Veitch said just seven cases had been diagnosed over the month and there had been 10 consecutive days of no new case diagnoses.
Coronavirus: All the latest updates on COVID-19 for Tasmania
"That is a very good position to be in, however, we do still have eight active cases and 15 close contacts in quarantine," he said.
Dr Veitch said it was disappointing testing numbers had declined over recent weeks.
"We need to be consistently having 500 tests each day to provide a level of reassurance that we are not missing coronavirus cases," he said.
Premier Peter Gutwein said a high level of test numbers would give Public Health Services and the government confidence to lift certain restrictions.
Using the analogy of a football match, he said the state was now playing the fourth quarter.
IN OTHER NEWS:
"But it ain't over until it's over," he said.
Mr Gutwein advised people to continue to follow the plan and observe current restrictions over the upcoming June weekend.
"Unlike other states, I'm not going to change our pathway because of political expediency," he said.
"We will review the restrictions next week as planned and move on from there."
Mr Gutwein said a worldwide drop in oil prices, associated with the coronavirus pandemic, and subsequent price decreases at the country's petrol bowsers had not been fully reflected in Tasmanian fuel prices.
He said the government's plan to introduce mandatory real-time fuel price reporting scheme was intended to put downward pressure on prices.
Mr Gutwein said if that did not result in the desired outcome, the government would consider price-capping if necessary.
Attorney-General Elise Archer said petrol retailers would be obliged to report prices in real-time under the scheme.
She said penalties would be applied if retailers failed to provide accurate information.
Ms Archer said it was expected the scheme would be operational by July to allow for consultation and development of a free mobile app for motorists to find the cheapest fuel prices.