Alexandersson–Haglund–Wang conjecture on Schur expansions of integral Jack polynomials

Let $\lambda$ and $\mu$ be partitions of $d$. Define the integral Jack polynomial by $J_\lambda(x;\tau)=c_\lambda(\tau)P_\lambda(x;\tau)$, and write its Schur expansion as

$$J_\lambda(\tau)=\sum_\mu v_{\lambda\mu}(\tau)s_\mu.$$

Define $a_k(\lambda,\mu)$ and $b_k(\lambda,\mu)$ by

$$v_{\lambda\mu}(\tau)=\sum_{k=0}^{d-1}a_k(\lambda,\mu)\binom{\tau+k}{d}=\sum_{k=1}^{d}b_{d-k}(\lambda,\mu)\binom{\tau}{k}k!.$$

Alexandersson–Haglund–Wang conjecture. The coefficients $a_k(\lambda,\mu)$ and $b_{d-k}(\lambda,\mu)$ are nonnegative integers. Moreover, the polynomials

$$\sum_{k=0}^{d}a_k(\lambda,\mu)z^k\qquad\text{and}\qquad\sum_{k=0}^{d}b_{d-k}(\lambda,\mu)z^k$$

have only real zeros. This conjecture concerns positivity, integrality, and real-rootedness of the Schur-expansion coefficients; the supplied text gives no resolution.