Extension conjecture for type B parking-space representations

Let $B_n$ be the signed symmetric group on $n$ letters, with inclusion $B_n\subset B_{n+1}$ given by choosing the elements of $B_{n+1}$ that fix the $(n+1)$st letter. For $m,n\geq 1$, let $\mathbb{C}[(\mathbb{Z}/m\mathbb{Z})^n]$ be the corresponding $B_n$-representation. Extension conjecture. For all $m,n\geq 1$, there exists a representation $V_{n,m}$ of $B_{n+1}$ such that

$$\operatorname{Res}^{B_{n+1}}_{B_n}V_{n,m}\cong \mathbb{C}[(\mathbb{Z}/m\mathbb{Z})^n].$$

This generalizes the extension question for the type $B$ parking space, which is the representation $\mathbb{C}[(\mathbb{Z}/(2n+1)\mathbb{Z})^n]$. The source presents this as its main conjecture; no resolution is given here.