Butler's Schur-positivity conjecture for Macdonald intersection polynomials

Let $\nu$ be a partition and let $\lambda,\mu\subseteq\nu$ be two distinct partitions such that $|\nu/\lambda|=|\nu/\mu|=1$. For a partition $\rho$, write

$$T_\rho:=\prod_{(i,j)\in\rho}t^{i-1}q^{j-1}.$$

Let $\widetilde{H}_\rho[X;q,t]$ denote the modified Macdonald polynomial, and define the Macdonald intersection polynomial by

$$\operatorname{I}_{\lambda,\mu}[X;q,t]=\frac{T_\lambda\widetilde{H}_\mu[X;q,t]-T_\mu\widetilde{H}_\lambda[X;q,t]}{T_\lambda-T_\mu}.$$

Butler's conjecture. The symmetric function $\operatorname{I}_{\lambda,\mu}[X;q,t]$ is Schur positive.

Butler's conjecture concerns positivity in the Schur basis for a divided difference of modified Macdonald polynomials. The paper gives a combinatorial positive monomial expansion and proves the conjecture in some special cases; no general resolution is stated here.