The central-commutator nilpotence conjecture for two endomorphisms

Let $V$ be a linear space over a field $\mathbbm{k}$ with $\operatorname{char}\mathbbm{k}=0$. Suppose $X,Y,Z\in\operatorname{End}(V)$ satisfy

$$Z=[X,Y],\qquad [X,Z]=[Y,Z]=0,$$

and let $v\in V$ and $p,q\in\mathbb{N}$ satisfy

$$X^{p+1}(v)=0,\qquad Y^{q+1}(v)=0.$$

Central-commutator nilpotence conjecture. Then

$$(X+Y)^{2p+2q+1}(v)=0.$$

This conjecture reduces the verification of a Serre relation in the relevant Drinfeld–Sokolov hierarchy construction to a nilpotence statement for endomorphisms whose commutator is central relative to both operators. The supplied context does not indicate whether the conjecture has been proved or disproved.