Generation of the representation ring by exterior powers for types B and odd D

Let $W$ be a Coxeter group of type $B_n$ or $D_{2n+1}$. Let $V$ be its reflection representation, and let $U$ be the non-faithful reflection representation of dimension $n-1$ described by the natural morphism $W\twoheadrightarrow\mathfrak{S}_n$. For each $k\geq 0$, write $\Lambda^k V$ and $\Lambda^k U$ for their $k$th exterior powers. Generation conjecture. The representation ring $R(W)$ is generated by

$$\Lambda^k V,\quad \Lambda^k U,\quad k\geq 0.$$

This is motivated by computer checks for small values of $n$ and extends the use of exterior powers of reflection representations to generate representation rings. For type $B_n$ and type $D_n$ with $n\geq4$, the exterior powers of the faithful reflection representation alone do not generate a subring of full rank; the conjecture proposes that adjoining the exterior powers of $U$ suffices.