Converse characterization of generating skew Hall–Littlewood functions

Let $\{\lambda^n/\mu^n\}$ be a graded skew partition sequence, and for each $n\geq0$ set $u_n=P_{\lambda^n/\mu^n}$. A skew diagram $\lambda^n/\mu^n$ is column-connected if it cannot be separated by a column: there is no $j$ such that the $j$th column has no cells while the diagram has cells both to the left and to the right of that column. Converse conjecture. If the set $\{u_n\}_{n\geq0}$ is algebraically independent and generates $\Lambda_{\mathbb{F}(t)}$, then, for every $n\geq0$, $\mu^n\subset\lambda^n$ and $\lambda^n/\mu^n$ is column-connected.

The preceding theorem proves that ribbon skew shapes give algebraically independent generating sets over $\mathbb{F}$. The converse is presented as an interesting open problem, concerning the characterization of skew Hall–Littlewood functions that generate the algebra of symmetric functions.