  We formulate a precise conjecture relating integral form partially-symmetric
Macdonald polynomials and the parabolic flag Hilbert schemes of Carlsson,
Gorsky, and Mellit. This extends, in an explicit fashion, Haiman's realization
of modified Macdonald symmetric functions via Hilbert schemes of points in the
plane. As evidence for our conjecture we prove that it is compatible with the
action of certain elements in Carlsson and Mellit's algebra $\mathbb{A}_{t,q}$,
including degree $1$ Pieri formulas.
