I don’t know if you’ll lose conversations, but you could try turning iMessages off in Settings ? Messages , waiting a several minutes, then turning it back on again.
I’m not brave enough to try this myself, since there are some pictures and history I want to keep.
Note: iMessages is Apples internet based message service. The Message app includes both Apple’s iMessage and cellular SMS services. The bubble is blue when sending via iMessage, and green when sending SMS texts.