The similarity characterization for products of two involutions

Let $V$ be an infinite-dimensional vector space, and let $u\in\operatorname{GL}(V)$. An involution is an invertible endomorphism whose square is the identity, and two endomorphisms are similar if they are conjugate by a vector-space isomorphism.

Involution-product conjecture. The endomorphism $u$ is the product of two involutions in $\operatorname{GL}(V)$ if and only if $u$ is similar to its inverse.

This conjecture extends the finite-dimensional theorem characterizing products of two involutions by similarity to the inverse. The infinite-dimensional setting has no invariant-factor criterion, so the validity of this equivalence remains open.