Codominant-support conjecture for parabolic Lusztig decomposition

Let $G=GL_n$, let $w$ be a codominant permutation, let $J=\\{n-k+1,\ldots,n-1\\}$, and let

$$f\colon G\times\mathcal{B}\longrightarrow G\times\mathcal{B}_J$$

be the forgetful map. Suppose the decomposition theorem gives

$$f_*(IC_{\mathcal{Y}_w})=\bigoplus_{z\in {}^J W}IC_{\mathcal{Y}_{z,J}}(L_{z,w}^{J',\emptyset}).$$

Codominant-support conjecture. The local system $L_{z,w}^{J',\emptyset}$ is zero for every non-codominant permutation $z$. The conjecture would restrict the decomposition-theorem summands to codominant strata in this important class of parabolic cases; the source presents it as suggested by examples.