McNamara's conjecture on cylindric Schur-positivity

Let $\lambda/d/\mu$ be a cylindric skew shape that is a subposet of $\mathfrak{C}_{k,n-k}$. For variables $x=(x_1,x_2,\ldots)$, call $s_{\lambda/d/\mu}(x)$ cylindric Schur-positive if it is a linear combination of cylindric Schur functions $s_{\nu/e/\emptyset}(x)$ with positive coefficients, where each $\nu/e/\emptyset$ is also a subposet of $\mathfrak{C}_{k,n-k}$. Cylindric Schur-positivity conjecture. Every cylindric skew Schur function is cylindric Schur-positive. This generalizes the Schur-positivity of ordinary skew Schur functions; the conjecture is known for cylindric ribbons, while the paper provides further evidence but does not establish it in general.