Uniform bound for simple linearly reductive Lie algebras in Verlinde categories

Let $p\geq5$, let ${\rm Ver}_p$ be the Verlinde category, and let $G$ be a simple linearly reductive height-$1$ group scheme in ${\rm Ver}_p$, with Lie algebra $\mathfrak{g}:={\rm Lie}G$. Write ${\rm FPdim}$ for Frobenius–Perron dimension. Uniform Lie-algebra bound conjecture. There exists a constant $D_p\geq1$ such that

$${\rm FPdim}\mathfrak{g}\le D_p$$

for every such $G$. The paper states that this conjecture implies the preceding linear bound for arbitrary linearly reductive height-$1$ subgroup schemes, after decomposing them into simple factors and copies of $(\mathbb{Z}/p)^\vee$.