Persistently high rates of COVID-19 have left the majority of the state’s nursing homes off limits to visitors, despite relaxed guidance meant to help reopen them.

Until this week, under state and federal rules, they could admit visitors only if they had no new infections among either patients or staff for 14 days.

That mark proved too hard for most to reach. A little more than half of the state’s 616 nursing homes were ineligible for indoor visits in mid-March, according to an Associated Press analysis of data from the U.S. Centers for Medicaid and Medicare. That’s the highest percentage of any state.

New York updated its visitation rules Thursday in a way that will now allow visits to resume under certain conditions, even if a resident or staffer has recently tested positive. But that relaxed standard might not clear the way for visitation in many homes having trouble keeping the virus out.