Commutant elements are commutators in regular Lie algebras

Let $L$ be a non-nilpotent regular Lie algebra with a nondegenerate symmetric invariant bilinear form. An element $x\in L$ is called a commutant element if $x$ belongs to the commutant of $L$, namely the derived subalgebra $[L,L]$. Commutant-commutator conjecture. Every commutant element of $L$ is a commutator: for each $x\in [L,L]$, there exist $a,b\in L$ such that

$$x=[a,b].$$

This extends the classical question of representing trace-zero matrices, and addresses the analogous problem for regular Lie algebras equipped with a nondegenerate symmetric invariant bilinear form. The source provides no resolution, so the conjecture remains open.