Does there exist a nontrivial finitely presented group in which every element is conjugate to its square? Is this an open problem?

Motivation: Jahren proved in [Geom Dedicata (2010)] that if $M$ is a closed manifold of dimension $\ge 5$ such that $\pi_1(M)$ has an element not conjugate to its square, then the smooth pseudoisotopy space for $M$ is not connected.

Finitely generated examples exist: Osin constructed a f.g. infinite torsion free group in which all nontrivial elements are conjugate.