We sailed Princess in the Caribbean in March, and there were no testing requirements, or immunization requirements either, if I remember correctly. So I suspect it is more a matter of the requirements of the destination countries being visited. I would suspect, though definitely don't know as a fact, that both Princess & HAL would end up with the same requirements for the same cruise destinations/times. You should be able to find their current requirements on their websites--although we had friends who were going to sail on Royal Caribbean in March transatlantic, and it was buried in the fine print that the transatlantic cruises required vaccines, even though everything on their main sites says they don't. They only found out a couple weeks before departure, so they ended up having to cancel last minute--thankfully, RCI gave them full credit.
How soon are you thinking of sailing?