The exchange-property characterization for sums of two square-zero endomorphisms

Let $V$ be an infinite-dimensional vector space, and let $u\in\operatorname{End}(V)$. An endomorphism $u$ has the exchange property if there is a decomposition $V=E\oplus F$ such that $u(E)\subseteq F$ and $u(F)\subseteq E$. Two endomorphisms are similar if they are conjugate by a vector-space isomorphism; the opposite of $u$ is $-u$.

Exchange-property conjecture. The endomorphism $u$ is the sum of two square-zero endomorphisms of $V$ if and only if $u$ has the exchange property. Moreover, if $\chi(\mathbb{F})\neq 2$, this condition is equivalent to $u$ being similar to its opposite.

This conjecture extends the finite-dimensional characterization of sums of two square-zero endomorphisms, where the exchange property is equivalent to the decomposition condition and, under the characteristic assumption, to similarity with the opposite. The infinite-dimensional case lacks invariant factors and remains open here.