'the' with the superlative superlative degree of adverbs

Hello.

Peter plays the piano better than Ann, but Mike plays ___/the best.
Peter runs faster than Ann, but Mike runs ___/the fastest.

Is ‘the’ to be used with the superlative superlative degree of adverbs or not?
Thanks in advance.

“the” is optional here.

Does it mean no agreement between grammarians?

I’m not aware that this is particularly an area of dispute. I think it’s just a case where two different forms are both accepted.