Stability conjecture for Ext between twisted two-part simple modules

Let $k$ be a field of characteristic $p>2$, let $\lambda,\mu\vdash d$ be partitions, and write $D^{p\lambda}$ and $D^{p\mu}$ for the corresponding simple modules of symmetric groups. Stability conjecture for twisted Ext. One has

$$\operatorname{Ext}^1_{k\Sigma_{pd}}(D^{p\lambda},D^{p\mu})\cong\operatorname{Ext}^1_{k\Sigma_{p^2d}}(D^{p^2\lambda},D^{p^2\mu}).$$

This predicts that the first extension group stabilizes after one additional twist. The statement is suggested by the established two-part case, but its validity for arbitrary partitions is not resolved here.