Goldberg–Jantzen type product formula for tempered generalized principal series

Let $I^G_P(\nu,\sigma)$ be a tempered generalized principal series, where $P=MN$ is a parabolic subgroup, $\sigma$ is a unitary supercuspidal representation of $M$, and $\nu\in\mathfrak{a}_M^*$. Let $\Phi_\sigma$ be the root system of roots $\alpha$ satisfying $w_\alpha.\sigma=\sigma$, decompose it into irreducible pieces $\Phi_\sigma=\sqcup_i\Phi_{\sigma,i}$, and let $M_i$ be the associated Levi subgroup. Assume that the associated $R$-group decomposes as $R_\sigma=\prod_iR_{\sigma,i}$ with $R_{\sigma,i}$ a subgroup of $W^{M_i}_M=N_{M_i}(M)/M$. Goldberg–Jantzen type product formula. There is a one-to-one correspondence

$$JH(I^G_P(\nu,\sigma))\leftrightarrows\prod_iJH(I^{M_i}_{M_i\cap P}(\nu,\sigma)).$$

This is presented as a conjectural product formula generalizing product formulas of Goldberg and Jantzen. The source gives it as a desired consequence of the proposed decomposition of the new $R$-group and does not provide a proof or resolution.