Positive key expansion conjecture for nonsymmetric Macdonald polynomials

Let $\alpha$ be a composition, let $\lambda=\operatorname{sort}(\alpha)$, and let $\gamma(P,\alpha)$ be a composition determined by $\alpha$ and $P$ that rearranges to the conjugate shape of $P$. Write $\mathrm{CoInvFree}(\alpha,w_0)$ for the coinversion-free fillings of shape $\alpha$ with decreasing basement $w_0$, and let $\mathcal{K}_{\gamma(P,\alpha)}(\mathbf{x})$ denote the corresponding key polynomial. Then

Positive key expansion conjecture.

$$\mathrm{E}_\alpha(\mathbf{x};q,0)=\sum_{F\in\mathrm{CoInvFree}(\alpha,w_0)}q^{\operatorname{maj}(F)}\mathbf{x}^F=\sum_{P\in\mathrm{SSYT}(\mu',\lambda')}q^{\operatorname{charge}(P)}\mathcal{K}_{\gamma(P,\alpha)}(\mathbf{x}).$$

In particular, if $S_Q$ is the subset of fillings in $\mathrm{CoInvFree}(\alpha,w_0)$ with recording tableau $Q\in\mathrm{SSYT}(\mu',\lambda')$, then $\sum_{F\in S_Q}\mathbf{x}^F$ is a key polynomial. This conjecturally extends the preceding partition-case charge formula to compositions and gives a positive key-polynomial expansion in the nonsymmetric setting.