Immanant positivity conjecture for Hankel matrices

Let $H=(h_{i,j})_{i,j\geq 0}$ be the Hankel matrix defined in the paper, and for $k\geq 1$ let $I=(i_1,\ldots,i_k)$ and $J=(j_1,\ldots,j_k)$ be sequences satisfying

$$0\leq i_1<\cdots<i_k,\qquad 0\leq j_1<\cdots<j_k.$$

Let $H_{I,J}$ denote the submatrix of $H$ with rows indexed by $I$ and columns indexed by $J$. For a partition $\lambda$ of $k$, write $\mathrm{Imm}_{\lambda}H_{I,J}$ for its immanant, and let $\geq_q$ denote coefficientwise nonnegativity in $q$. Immanant positivity conjecture. For every such $k$, $I$, $J$, and partition $\lambda$ of $k$,

$$\mathrm{Imm}_{\lambda}H_{I,J}\geq_q 0.$$

This extends the previously proved immanant positivity for a large family of Catalan–Stieltjes matrices and their associated Hankel matrices. It predicts coefficientwise immanant positivity for all minors of the Hankel matrix $H$, generalizing determinant-based total positivity; the paper presents it as a conjecture, with no resolution supplied here.