The affine W-algebra action conjecture for non-simply laced affine Lie algebras

Let $\cal G$ be an affine Lie algebra with connected Dynkin diagram, let $\cal G^{\vee}$ be its Langlands dual, and let $G_{\cal G^{\vee}}$ be the semisimple simply connected group obtained by removing the affine vertex from the Dynkin diagram of $\cal G^{\vee}$. Let $\cal U_{\cal G^{\vee}}^d$ be the corresponding Uhlenbeck spaces, equipped with the action of $G_{\cal G^{\vee}}\times \mathbb C^*\times\mathbb C^*$. Affine W-algebra action conjecture. There exists an action of $\mathscr W(\cal G)$ on

$$\bigoplus_d \operatorname{IH}^*_{G_{\cal G^{\vee}}\times \mathbb C^*\times \mathbb C^*}(\mathcal U_{\cal G^{\vee}}^d)$$

with properties similar to those of the main theorem for simply laced groups. This would extend the geometric realization of W-algebras from untwisted and simply laced settings to arbitrary affine Lie algebras, including twisted cases; the source postpones the construction and does not establish the asserted action.