Classification of rank-two simple Lie algebras in Verlinde categories

Let $p>0$, let ${\rm Ver}_p^+$ be the relevant Verlinde category, and let $\mathfrak{g}$ be a simple Lie algebra in ${\rm Ver}_p^+$ containing a simple Lie algebra $\mathfrak{s}$ such that $\mathfrak{s}$ acts on $\mathfrak{g}$ canonically. Define the rank of $\mathfrak{g}$ to be its length as an $\mathfrak{s}$-module, equivalently as an object of ${\rm Ver}_p^+$. Rank-two classification conjecture. There are no simple Lie algebras of rank $2$ in ${\rm Ver}_p^+$ other than the listed series and exceptional examples: $A_2$ for $p\geq7$, $B_2=C_2$ for $p\geq11$, $D_2^*$ for $p\geq11$, $G_2$ for $p\geq17$, and the exceptional cases $E_2^*$ at $p=23$ and $E_2^{**}$ at $p=37$. This is a proposed classification of the rank-two simple Lie algebras occurring in the Verlinde category.