Rapid generation conjecture for finite simple Lie algebras

Let $\operatorname{\mathfrak{g}}$ be a Lie algebra over $\mathbf{F}_p$. For subsets $X,Y\subseteq \operatorname{\mathfrak{g}}$, define

$$X+Y=\{x+y\mid x\in X,\ y\in Y\},\qquad [X,Y]=\{[x,y]\mid x\in X,\ y\in Y\}.$$

For a generating set $A$ of $\operatorname{\mathfrak{g}}$, define $A^1=\{0\}\cup A$ and, for $k\geq 2$,

$$A^k=\bigcup_{0<j<k}\bigl((A^j+A^{k-j})\cup[A^j,A^{k-j}]\bigr).$$

The diameter $\operatorname{diam}(\operatorname{\mathfrak{g}},A)$ is the least $k$ such that $A^k=\operatorname{\mathfrak{g}}$.

Rapid generation conjecture. There exist absolute constants $C,D$ such that, for every nonabelian finite simple Lie algebra $\operatorname{\mathfrak{g}}$ over $\mathbf{F}_p$ and every generating set $A$ of $\operatorname{\mathfrak{g}}$,

$$\operatorname{diam}(\operatorname{\mathfrak{g}},A)\leq C(\log|\operatorname{\mathfrak{g}}|)^D.$$

This is presented as the linear analogue of Babai's conjecture for finite simple groups. The supplied text does not establish whether it is open or resolved; the paper's surrounding results provide explicit diameter bounds but do not, in the supplied span, identify this exact assertion as proved.