Positive Hall–Littlewood expansion in the immaculate basis

Let $\lambda$ be a partition, let $\mathcal{Q}'_\lambda$ be the lifted Hall–Littlewood basis element indexed by $\lambda$, and let $\mathfrak S_\beta$ denote the immaculate basis. For an immaculate tableau $T$, write $content(T)$ for its content and $shape(T)$ for its shape.

Hall–Littlewood immaculate positivity conjecture. The element $\mathcal{Q}'_\lambda$ expands in the immaculate basis with coefficients that are positive polynomials in $q$. More explicitly,

$$\mathcal{Q}'_\lambda=\sum_T q^{st(T)}\mathfrak S_{shape(T)},$$

where $st$ is a statistic and the sum is over all immaculate tableaux of content $\lambda$.

This is proposed as an analogue of the positive Schur expansion of Hall–Littlewood functions. The statistic $st$ is not defined in the source, and the conjecture remains unresolved in the supplied text.