TITLE: Finite presentability and elementary equivalence
QUESTION [18 upvotes]: Do there exist two elementary equivalent finitely generated groups $G,H$ such that $G$ is finitely presented but $H$ is not finitely presentable?

It seems reasonable to think that finite presentability is not preserved under elementary equivalence among finitely generated groups, so such groups probably exist. However, I do not know any example. The case where $G,H$ have the same universal theory, instead of having the same first order theory, would be already interesting.

REPLY [2 votes]: The problem is known and still open but does not seem hopeless. I would start with considering Zilber's old example of two f.g. non-isomorphic nilpotent of class $2$ groups which are e.e.: B.I. Zilʹber,
An example of two elementarily equivalent but not isomorphic finitely generated metabelian groups. Algebra i Logika 10 (1971), 309–315. Both of his groups were of course finitely presented. But I would guess that there are similar examples when one of the groups is  not finitely presented.