Macdonald superpolynomial Hall–Littlewood positivity conjecture

Let $P_\Lambda(x,\theta;q,t)$ be the Macdonald superpolynomials indexed by superpartitions, and let $s_\Lambda(x,\theta)$ and $\bar s_\Lambda(x,\theta)$ be the two Schur-superpolynomial limits. Define coefficients by

$$s_\Lambda=\sum_{\Omega\leq\Lambda}\bar K_{\Lambda\Omega}m_\Omega,\qquad \bar s_\Lambda=\sum_{\Omega\leq\Lambda}K_{\Lambda\Omega}m_\Omega.$$

Hall–Littlewood positivity conjecture. The coefficients $K_{\Lambda\Omega}(t)$ and $\bar K_{\Lambda\Omega}(t)$ defined by

$$s_\Lambda=\sum_{\Omega\leq\Lambda}\bar K_{\Lambda\Omega}(t)P_\Omega(t),\qquad \bar s_\Lambda=\sum_{\Omega\leq\Lambda}K_{\Lambda\Omega}(t)\bar P_\Omega(1/t)$$

are polynomials in $t$ with nonnegative integer coefficients. Moreover, $\bar K_{\Lambda\Omega}(1)=\bar K_{\Lambda\Omega}$ and $K_{\Lambda\Omega}(1)=K_{\Lambda\Omega}$. The conjecture proposes a positive change-of-basis theory between Schur and Hall–Littlewood superpolynomials, extending the ordinary theory to superspace.