Galashin–Lam conjecture on $q,t$-Schur positivity for ${\mathbb Z}$-convex curves

A curve $C$ passing through lattice points $(0,0)={\bm p}_0,{\bm p}_1,\ldots,{\bm p}_k=(m,n)$ is ${\mathbb Z}$-convex if the set of lattice points strictly above $C$, together with the points ${\bm p}_i+(-\epsilon,\epsilon)$ for sufficiently small $\epsilon>0$, is ${\mathbb Z}$-convex. Write $\operatorname{k}(C)=k$ and let $F_C$ be the symmetric-function invariant associated to $C$. Galashin–Lam's $q,t$-Schur-positivity conjecture. For every ${\mathbb Z}$-convex curve $C$, the formal power series

$$\frac{1}{(1-t)^{\operatorname{k}(C)-1}}F_C$$

is $q,t$-Schur positive. Here $(1-t)^{-1}=1+t+t^2+\cdots$ is interpreted as a formal power series. This extends the cited conjecture of Blasiak, Haiman, Morse, Pun, and Seelinger; the source gives no resolution.