Schur and shifted-$e$ positivity conjecture for generalized LLT polynomials

Let $w\in S_n$ and define the generalized LLT polynomial by

$$\operatorname{LLT}(w):=(q-1)^n\left(\omega\left(\operatorname{ch}\left(q^{\frac{\ell(w)}{2}}C'_w\right)\right)\right)\left[\frac{\mathbf{x}}{q-1}\right],$$

where $C'_w$ is a Kazhdan–Lusztig element and $\omega$ is the standard involution on symmetric functions. A symmetric function is shifted $e$-positive if it is a nonnegative combination in the shifted elementary basis. Generalized LLT positivity conjecture. For every $w\in S_n$, $\operatorname{LLT}(w)$ is Schur-positive and shifted $e$-positive. The source motivates this by the corresponding known properties for codominant permutations and gives no resolution.