A few more months isn't going to give us a vaccine so at worst that's just causing more economic damage, leaving hospitals empty and moving the pandemic two months into the future. At best you open back up with safety procedures, and limit the spread. But there's no reason you can't do that exact same thing today.
Also, I agree with Musk that an indefinite lockdown is unconstitutional. And if it's not, I want our laws changed so it is. It may be a good idea health-wise (or may not, see increased suicides) but that doesn't make it legal and letting an unelected official impose an indefinite lockdown is just bad policy. So if the local authority couldn't give a deadline, I'd tell them to piss off too.