The restricted cohomological-dimension-one structure conjecture for Lie $p$-algebras

Let $L$ be a Lie $p$-algebra, and let $\operatorname{cd}_*(L)$ denote its restricted cohomological dimension. Let $\mathscr L$ be a free Lie $p$-algebra, let $T_1,\dots,T_n$ be finite-dimensional tori, and let each symbol $\bowtie$ denote either $\rtimes$, the action of the left-hand side on the right-hand side, or $\ltimes$, the action of the right-hand side on the left-hand side. Thus an algebra of the displayed form is

$$(\dots ((\mathscr L \bowtie T_1) \bowtie T_2) \dots ) \bowtie T_n.$$

Restricted cohomological-dimension-one structure conjecture. Any Lie $p$-algebra $L$ with $\operatorname{cd}_*(L)=1$ is of the form

$$(\dots ((\mathscr L \bowtie T_1) \bowtie T_2) \dots ) \bowtie T_n.$$

The claim proposes a structural classification of Lie $p$-algebras of restricted cohomological dimension one by iterated extensions of a free Lie $p$-algebra by finite-dimensional tori. The supplied context gives the construction and motivation but does not state a resolution, so the conjecture is recorded as open.