Equivariant degree-one injectivity for braid-matroid Varchenko–Gelfand duals

Let $cmathcal{M}=cmathrm{Br}_n$ be the braid oriented matroid, and let $cmathrm{VG}(cmathrm{Br}_n)^!_i$ denote the degree-$i$ component of its Koszul dual. Let $cmathcal{S}_{cmathrm{VG}}(m,n-1)$ denote the corresponding $cmathfrak{S}_n$-representation.

Braid-matroid equivariant injectivity conjecture. For all $i eggeq 1$, there exist equivariant injections

$$[cmathcal{S}_{cmathrm{VG}}((n-1)+i,n-1)]=[cmathrm{VG}(cmathrm{Br}_n)^!_i] \hookrightarrow [cmathcal{S}_{cmathrm{VG}}((n-1)+i+1,n-1)]=[cmathrm{VG}(cmathrm{Br}_n)^!_{i+1}].$$

Known degree-two injections exist generally for oriented matroids, but degree-one equivariant injections need not. Computations for $n egleq 10$ and $1 egleq i egleq 9$ support this braid-matroid claim; a proof is not supplied.