Joint-growth height conjecture for wreath and Kronecker products

Let $S_n$ and $S_m$ be symmetric groups, let $\pi$ be distributed uniformly from either the wreath product $S_n \wr S_m$ or the Kronecker product $S_m \otimes S_n$, and let $h_{n,m}$ denote the height of the associated block binary search tree. Write $\to_p$ for convergence in probability.

Joint-growth height conjecture. As $\min(n,m)\to\infty$,

$$\frac{h_{n,m}}{\log n\\,\log m}\to_p c^*.$$

Here $c^*\approx 4.311$ is the constant from Devroye's theorem. This is motivated by the fixed-$n$ conjecture and the asymptotic relation $H_n^{(1)}/\log n\to 1$; proving the matching result when both parameters grow remains open and would require stronger control of tree-height extremes.