Haiman–Hugland k-Schur positivity conjecture for LLT polynomials

Let $\bm{\lambda}=(\lambda^{(0)},\lambda^{(1)},\ldots,\lambda^{(d-1)})$ be a $d$-tuple of skew partitions, and let $c(x)$ denote the usual content of a cell $x$ in its component. Let $\omega$ be the conjugate map on the symmetric-function ring, and let $p$ be an integer and $k>0$ such that every cell $x$ of $\bm{\lambda}$ satisfies

$$p\leq c(x)<p+k.$$

Haiman–Hugland's conjecture. The symmetric function

$$\omega\tilde{G}_{\bm{\lambda}}[X;q]$$

is $k$-Schur positive.

This is a strengthening of the known Schur positivity of LLT polynomials, asserting positivity in the more refined $k$-Schur basis for tuples of skew partitions supported on $k$ adjacent diagonals. The source presents this as a conjecture of Haiman and Haglund; its resolution status is not specified here.