Stanley's positivity conjecture for Jack character polynomials on multirectangular partitions

Let {\text{\boldmath p}}=(p_1,\ldots,p_m) and {\text{\boldmath q}}=(q_1,\ldots,q_m) be sequences of positive integers with $q_1\geq\cdots\geq q_m$, and let {\text{\boldmath p}}\times{\text{\boldmath q}} be the partition formed by the union of rectangles of sizes $p_i\times q_i$. Let \lambda={\text{\boldmath p}}\times{\text{\boldmath q}}, let $\mu$ be a partition with $m_1(\mu)=0$ and $|\mu|=k\leq|\lambda|=n$, let $\beta=\alpha-1$, and write $\mu=(1^{m_1(\mu)},2^{m_2(\mu)},\ldots)$ and $z_\mu=\prod_{i\geq1}i^{m_i(\mu)}m_i(\mu)!$. The Jack polynomials are expanded as $J_\lambda(\alpha)=\sum_{|\rho|=|\lambda|}\theta^\lambda_\rho(\alpha)p_\rho$. Stanley's positivity conjecture. The quantity $z_\mu\theta^\lambda_{\mu,1^{n-k}}(\alpha)$ is a polynomial in ({\text{\boldmath p}},{\text{\boldmath q}},\beta) with integer coefficients; after replacing {\text{\boldmath q}} by -{\text{\boldmath q}}, the coefficients of $(-1)^kz_\mu\theta^\lambda_{\mu,1^{n-k}}(\alpha)$ are nonnegative integers, and at least one coefficient is $1$. This extends the corresponding positivity theorem for normalized symmetric-group characters and is presented as a conjectured Jack-polynomial analogue; the source gives no resolution of the full multirectangular claim.