The classical-and-abelian composition-factor conjecture for the modular Plesken Lie algebra

Let $\mathrm{G}$ be a finite group, let $p$ be a modular prime for $\mathrm{G}$, and let $\mathcal{L}_p[\mathrm{G}]$ be the associated modular Plesken Lie algebra. Let $k[\mathrm{G}]=b_1\oplus\cdots\oplus b_r$ be its block decomposition, and let $\mathcal{X}_1$, $\mathcal{X}_{-1}$, and $\mathcal{X}_0$ be the subsets of Brauer characters classified by Frobenius–Schur indicator. For each Brauer character $\phi$, write $\phi(1)$ for its degree, and use $\mathfrak{o}_{\phi(1)}$, $\mathfrak{sp}_{\phi(1)}$, and $\mathfrak{gl}_{\phi(1)}$ for the corresponding classical Lie algebras; when $p\mid\phi(1)$, let $\mathfrak{PA}_{\phi(1)-1}$ denote the codimension-1 simple factor of $\mathfrak{A}_{\phi(1)-1}$. The classical-and-abelian composition-factor conjecture. The Lie algebra $\mathcal{L}_p[\mathrm{G}]$ admits a direct sum decomposition

$$\mathcal{L}_p[\mathrm{G}]=\bigoplus_{j=1}^r\mathfrak{a}_j,$$

where $\mathfrak{a}_j$ is the projection onto $b_j$ and need not be simple; its composition factors are either abelian or of classical type. More precisely, each $\phi\in\mathcal{X}_1$ gives a composition factor isomorphic to $\mathfrak{o}_{\phi(1)}$, each $\phi\in\mathcal{X}_{-1}$ gives one isomorphic to $\mathfrak{sp}_{\phi(1)}$, and each pair of conjugate Brauer characters in $\mathcal{X}_0$ gives one isomorphic to $\mathfrak{gl}_{\phi(1)}$, whose codimension-1 simple factor is $\mathfrak{sl}_{\phi(1)}$ unless $p\mid\phi(1)$, in which case it has a codimension-2 simple factor of type $\mathfrak{PA}_{\phi(1)-1}$. In addition, a defect-zero block of $k[\mathrm{G}]$ gives rise to a direct summand of $\mathcal{L}[\mathrm{G}]$ of classical type. The conjecture excludes exceptional modular Lie algebras as simple composition factors; the stated decomposition and the correspondence with Brauer-character degrees and indicators describe the expected modular analogue of the complex case, while the abelian remaining factors distinguish the modular setting.