The shuffle-algebra image conjecture for the preprojective CoHA

Let $Q$ be a quiver, let $T$ be a torus as in Definition, let $\iota_*$ be the map into the localized shuffle realization, and let $\mathcal{S}^+$ be the shuffle algebra of symmetric polynomials satisfying the wheel conditions. Then

Shuffle-algebra image conjecture.

$$\operatorname{Im}\iota_* = \mathcal{S}^+.$$

This conjecture would give a full description of the image of the localized CoHA in terms of wheel conditions. The source has already established the inclusion $\operatorname{Im}\iota_*\subseteq\mathcal{S}^+$, while equality remains conjectural there.