The polynomial-positive codominant character decomposition conjecture

Let $S_n$ be the symmetric group, let $C'_w$ denote the Kazhdan–Lusztig basis element associated to $w\in S_n$, and let $\operatorname{ch}$ denote the dual Frobenius character. Call a permutation codominant according to the paper's definition, and let $\mathbb{N}[q]$ be the polynomials in $q$ with non-negative integer coefficients.

Polynomial-positive decomposition conjecture. For each permutation $w\in S_n$, there exist codominant permutations $w_1,\ldots,w_k\in S_n$ such that $\operatorname{ch}(q^{\frac{\ell(w)}{2}}C'_w)$ is a combination of $\operatorname{ch}(q^{\frac{\ell(w_i)}{2}}C'_{w_i})$ with coefficients in $\mathbb{N}[q]$.

The paper presents this as a weaker version of Haiman's disproved decomposition conjecture. The supplied text gives no resolution of this weaker conjecture.