Refined h-positivity conjecture for equivalence classes of heaps

Let $P$ be a natural unit interval order on $[n]$, let $\mu\in\mathbb{N}^n$, and let $\mathcal{H}(P,\mu)/\sim$ be the set of equivalence classes of heaps. For an equivalence class $[H]$, write

$$K_{[H]}(\mathbf{x})=\sum_{H'\in[H]}K_{H'}(\mathbf{x}).$$

Here $h$-positive means having a nonnegative expansion in the complete homogeneous symmetric-function basis, and $e$-positive means having a nonnegative expansion in the elementary basis. Refined h-positivity conjecture. For every $[H]\in\mathcal{H}(P,\mu)/\sim$, $K_{[H]}(\mathbf{x})$ is $h$-positive. In particular, $X_P(\mathbf{x},q;\mu)$ is $e$-positive. This refines the $e$-positivity conjecture by seeking positivity for each equivalence-class contribution; the source gives no resolution.