Conjecture on indecomposable summands of twisted Foulkes modules

Let $H^{(2^m;k)}$ denote the twisted Foulkes module for the symmetric group $S_{2m+k}$. Let $\tau$ be the $p$-content and $\theta$ the odd sequence of a partition of $2m+k$, with $k=|\theta|$. For a partition label $\tau-\theta$, write $H^{(2^m)}_{\tau-\theta}$ for the corresponding summand of the Foulkes module, and let $\boxtimes$ denote the outer tensor product, $\big\uparrow^{S_{2m+k}}$ induction to $S_{2m+k}$, and $(]_{\tau}$ projection to the block or $p$-content $\tau$. Twisted Foulkes indecomposability conjecture. For all integers $m,k\geq 0$,

$$\left(\left(H^{(2^m)}_{\tau-\theta}\boxtimes\operatorname{sgn}\right)\big\uparrow^{S_{2m+k}}\right)_{\tau}$$

is indecomposable. Equivalently,

$$H^{(2^m;k)}=\bigoplus_{(\tau,\theta)}\left(\left(H^{(2^m)}_{\tau-\theta}\boxtimes\operatorname{sgn}\right)\big\uparrow^{S_{2m+k}}\right)_{\tau},$$

where the sum runs over all pairs $(\tau,\theta)$ arising from a partition of $2m+k$ with $\tau$ its $p$-content and $\theta$ its odd sequence. This conjecture would give the decomposition of every twisted Foulkes module into indecomposable summands, extending the established decomposition for ordinary Foulkes modules. The source provides no resolution status.