Factorizability conjecture for Stanley Jack $g$-polynomials

Let $bbig \lambda$, $bbig mu$, and $bbig nu$ be partitions, and let $c^bbig \lambda_{bbig mu,bbig nu}$ be the Littlewood–Richardson coefficient. For each box $s$ of a partition $bbig pi$, let $c_{bbig pi,s}$ and $c'_{bbig pi,s}$ denote the two Jack hook factors, and write $g^bbig \lambda_{bbig mu,bbig nu}$ for the corresponding Jack $g$-polynomial.

Factorizability conjecture. If

$$c^\lambda_{\mu,\nu}=1,$$

then $g^bbig \lambda_{bbig mu,bbig nu}$ is a product of linear factors. More precisely,

$$g^\lambda_{\mu,\nu}=\left(\prod_{s\in\lambda}c^*_{\lambda,s}\right)\left(\prod_{s\in\mu}c^*_{\mu,s}\right)\left(\prod_{s\in\nu}c^*_{\nu,s}\right),$$

where, for $bbig pi=bbig \lambda,bbig mu,bbig nu$ and $s\inbbig pi$, either $c^*_{bbig pi,s}=c_{bbig pi,s}$ or $c^*_{bbig pi,s}=c'_{bbig pi,s}$, with each of the two choices occurring $|bbig \lambda|$ times in total.

The conjecture concerns a particularly rigid factorization pattern for Jack structure polynomials when the relevant Littlewood–Richardson coefficient is one. The source proves it in several special cases, but leaves the general assertion open.