Take the MG's out and toss them. If they have seed pods mature enough, they will dry and spread seed.
Japanese Morning Glories are not as rampant at re-seeding as others. HOWEVER, they can cross polinate and become more like the rampant reseeders...so I grow a select few MG's, and grow them in places where I can easily take off dry seed pods and where I am reminded to do this daily.

I do not have a problem with over reseeding this way.

MGs are annuals. So, if you pull them out before their seeds mature, they will be finished. Their seed pods stay intact on vine for a good while and you will have a chance to pick them, keep them or destroy them.

Lets suppose that somehow some seeds escape and grow next year. Do not fear! it is easy to pull them up as we do with tens of weeds all the time.

I let some wild MGs to grow between my tomatoes. It is beautiful. they do not harm tomatoes even if they winde around tomato branches.
You can trim then if they go beyond where you don't want them to.

Thanks. I'll keep pulling.