A solution to Butler's positivity conjecture

Let $\lambda ,\mu ,\nu$ be distinct partitions such that $\lambda ,\mu \subset \nu$ and $\mathrm{\mid }\nu \mathrm{/}\lambda \mathrm{\mid }=\mathrm{\mid }\nu \mathrm{/}\mu \mathrm{\mid }=1$. We prove Butler’s positivity conjecture posed in 1994: the expansion of the Macdonald intersection polynomial $$\frac{T_{\lambda }{\tilde{H}}_{\mu }(X;q,t)-T_{\mu }{\tilde{H}}_{\lambda }(X;q,t)}{T_{\lambda }-T_{\mu }}$$ in terms of the Schur function basis has coefficients in ${\mathbb{Z}}_{\ge 0}[q,t]$.