Parabolic branching polynomial positivity conjecture

Let $\mathfrak{g}$ be a semisimple Lie algebra with Weyl group $W$, positive roots $R_{+}$, weight lattice $P$, and dominant weights $P_{+}$. Let $\overline{\mathfrak{g}}$ be a parabolic subalgebra with dominant weights $\overline{P}_{+}$, and let $\overline{\lambda}+\gamma\in\overline{P}_{+}$. For a dominant weight $\nu\in P_{+}$, define

$$b_{\nu,\overline{\lambda}+\gamma}(p)=\sum_{w\in W}\varepsilon(w)\mathrm{\hat{P}}_{p}(w(\nu+\rho)-\overline{\lambda}-\gamma-\rho),$$

where $\mathrm{\hat{P}}_{p}$ is the partition function determined by

$$\prod_{\alpha\in R_{+}\setminus\overline{R}_{+}}\frac{1}{1-pe^{\alpha}}=\sum_{\eta\in\hat{Q}_{+}}\mathrm{\hat{P}}_{p}(\eta)e^{\eta}.$$

Parabolic branching polynomial positivity conjecture. The polynomial $b_{\nu,\overline{\lambda}+\gamma}(p)$ has nonnegative integer coefficients at least when $\overline{\lambda}+\gamma\in P_{+}$, that is when $\overline{\lambda}+\gamma$ is a dominant weight for $\mathfrak{g}$ (and not only for $\overline{\mathfrak{g}}$). This polynomial specializes at $p=1$ to the multiplicity of $\overline{V}(\overline{\lambda}+\gamma)$ in the restriction of $V(\nu)$ from $\mathfrak{g}$ to $\overline{\mathfrak{g}}$. The conjecture is only partially proved according to the source, including a result of Broer in type $A$; the stated positivity in the indicated dominant-weight case remains open.