I would say no. I can't give a scientifically referenced answer but I never harvest all the stalks and all the sources I've ever read say to let it die down naturally. I stop picking in July because we just get bored with it and it needs to build up its roots. Also it gets tough and stringy as it gets old.

I was always told to stop around July 4th so the plant had time to rejuvenate itself for next years harvest.

Agree that one should never harvest all of it. Removing all or even most of its leaves markedly reduces its ability to photosynthisize energy needed for good root development. And like the others have said, I stop harvesting in mid-July for health of the plant.

Dave