Left-right symmetry conjecture for affine involution Stanley symmetric functions

Let $n$ be a positive integer, let $\tilde I_n$ denote the set of affine involutions, let $z\in\tilde I_n$, let $\hat{F}_z$ be the affine involution Stanley symmetric function, let $\omega^+$ be the relevant involution on symmetric functions, let $\mathcal{A}(z)$ be the set of affine atoms associated with $z$, and let $F_\pi$ denote the affine Stanley symmetric function indexed by $\pi$. Left-right symmetry conjecture. For every $z\in\tilde I_n$,

$$\omega^+(\hat{F}_z)=\hat{F}_z,$$

that is,

$$\sum_{\pi\in\mathcal{A}(z)}F_{\pi^{-1}}=\sum_{\pi\in\mathcal{A}(z)}F_\pi.$$

This conjecture says that the left- and right-handed conventions yield the same affine involution Stanley symmetric function, as suggested by the corresponding left-handed versions of the coproduct and comodule results. The statement is supported by computations, but remains open in the source.