Cartan-type non-maximal-height bound conjecture

Let $\mathfrak{X}(n,\underline{m})$ denote the minimal $p$-envelope of a simple Cartan-type Lie algebra $\mathsf{X}(n,\underline m)^{(2)}$, where $\mathsf{X}\in\{\mathsf{W},\mathsf{S},\mathsf{H},\mathsf{K}\}$. Let $\chi$ be a character of non-maximal height. Cartan-type non-maximal-height conjecture. There are at most

$$p^{\mathsf{mt}(\mathsf{X}(n,\underline{1}))}$$

nonisomorphic simple $\mathsf{u}(\mathfrak{X}(n,\underline{m}),\chi)$-modules. The source presents this as a strengthening of part (i) of the preceding conjecture; it notes that the preceding bound is known for some minimal $p$-envelopes, but gives no general proof of this strengthening.