Linear bound for Lie algebras of linearly reductive height-one group schemes

Let $p\geq5$, let ${\rm Ver}_p$ be the Verlinde category, and let $G$ be a linearly reductive height-$1$ subgroup scheme of $GL(Y)$ for an object $Y\in {\rm Ver}_p$. Let $\mathfrak{g}:={\rm Lie}G$, and write ${\rm FPdim}$ for Frobenius–Perron dimension. Group-scheme Lie-algebra conjecture. There exists a constant $M_p>0$ such that

$${\rm FPdim}\mathfrak{g}\le M_p{\rm FPdim}Y.$$

The paper explains that this conjecture would imply the asymptotic linear bound for tensor powers. It is stronger than the quadratic bound coming from $\mathfrak{g}\subseteq\mathfrak{gl}(Y)$ and is restricted to linearly reductive height-$1$ group schemes.