We show that the Frobenius character of the equivariant Borel-Moore homology of a certain positive $GL_n$-version of the unramified affine Springer fiber $Z_k$ studied by Goreski, Kottwitz and MacPherson is computed by the matrix coefficients of the $\nabla^k$-operator, which acts diagonally in the modified Macdonald basis. We do this by relating the combinatorial formula for the $\nabla^k$-operator we obtained in an earlier paper to the GKM paving of $Z_k$, and we give an algebraic presentation of the above homology as an explicit submodule of the Kostant-Kumar nil Hecke algebra. We then study a certain open locus $U_k \subset Z_k$, and reduce a long-standing conjecture of Bergeron, Garsia, Haiman and Tesler, which predicts the sign of the coefficients of the Schur expansion of $\nabla^k$, to a vanishing conjecture about the homology groups of $U_k$. The latter conjecture is in turn reduced to a vanishing conjecture for certain open loci of the regular semisimple Hessenberg varieties which are indexed by partial Dyck paths.