Componentwise p-positivity conjecture for ribbon Schur Q-functions

Let $D=D_1\cup\cdots\cup D_r$ be a ribbon with connected components $D_i$, for $i=1,\dots,r$, and let $\mathfrak{r}_D$ denote its ribbon Schur $Q$-function. Componentwise p-positivity conjecture. Then $\mathfrak{r}_D$ is $p$-positive if and only if $\mathfrak{r}_{D_i}$ is $p$-positive for every $i$. Products of $p$-positive symmetric functions show one direction, but the converse is not true in general for arbitrary products and remains open for ribbons; the conjecture was checked computationally in specific cases.