Bergeron's combinatorial supersymmetry conjecture

Let $\mathcal{R}(n;r,\ell)$ be the multigraded $S_n$-module with commuting and anticommuting variable sets, let $\operatorname{ch}\mathcal{R}(n;r,\ell)$ be its $S_n\times GL_r(\mathbb{C})\times GL_\ell(\mathbb{C})$ character, and let

$$\mathcal{E}_n(\mathbf{x},\mathbf{q})=\sum_{\lambda,\mu}b_{\lambda,\mu}s_\lambda(\mathbf{x})s_\mu(\mathbf{q})$$

be the stable character for commuting variables. Let $\mathbf{z}$ be an infinite alphabet of anticommuting tracking variables and $c_{\nu,\rho}^{\mu}$ a Littlewood–Richardson coefficient. Bergeron's combinatorial supersymmetry conjecture. For any $r,\ell\geq0$, $\operatorname{ch}\mathcal{R}(n;r,\ell)$ is obtained from

$$\sum_{\lambda,\mu,\nu,\rho}b_{\lambda,\mu}c_{\nu,\rho}^{\mu}s_\lambda(\mathbf{x})s_\nu(\mathbf{q})s_{\rho'}(\mathbf{z})$$

by setting $q_i=0$ for $i>r$ and $z_j=0$ for $j>\ell$. The conjecture predicts that all mixed commuting–anticommuting characters are determined by the stable commuting character. The source gives no resolution status.