Nekrasov's wheel-condition generation conjecture for tripled quivers

Let $Q$ be a quiver, let $\tilde{Q}$ be its triple quiver with canonical cubic potential $\tilde{W}$, and let $T$ be the torus of Example corresponding to the strong assumption, with coefficient ring $R$ and fraction field $\operatorname{Frac}(R)$. Let $\mathcal{S}$ be the wheel-condition subalgebra of the shuffle algebra, let $\mathring{\mathcal{S}}$ be its spherical subalgebra, and write

$$\mathcal{S}_{\mathrm{loc}}=\mathcal{S}\otimes_R\operatorname{Frac}(R),\qquad \mathring{\mathcal{S}}_{\mathrm{loc}}=\mathring{\mathcal{S}}\otimes_R\operatorname{Frac}(R).$$

Nekrasov's wheel-condition generation conjecture. The localized wheel-condition algebra equals the localized spherical subalgebra:

$$\mathcal{S}_{\mathrm{loc}}=\mathring{\mathcal{S}}_{\mathrm{loc}}.$$

This is the cohomological analogue of the corresponding $K$-theoretic wheel-condition generation statement. It is used to identify the localized CoHA with the full wheel-condition algebra, but the source does not provide a resolution.