Betti-origin deformation conjecture for the annular Hecke functor

Let $\sf{H}_W$ be the Hecke category, let $\sf{AH}:\sf{H}_W^{\operatorname{op}}\to\mathsf{Mod}_2^b(\bb{A}_W)$ be the annular Hecke functor, and let $\bb{A}_{W,\epsilon}$ and $\bb{D}_{\nu,\epsilon}^{\operatorname{rat}}$ denote the deformed algebras. For periodic $\beta\in\mathit{Br}_W^+$ of regular elliptic slope $\nu$, let $L_\nu(1)$ be the simple rational DAHA module and let $\bm{f}_{\leq *}$ be its filtration. Betti-origin deformation conjecture. There is a flat deformation of $\sf{AH}$ over $\overline{\mathbf{Q}}_\ell[\epsilon]$ to a functor

$$\sf{AH}_\epsilon:\sf{H}_W^{\operatorname{op}}\to\mathsf{Mod}_2^b(\bb{A}_{W,\epsilon})$$

such that the $\bb{A}_W$-action extends to a weight-filtered $\bb{A}_{W,\epsilon}$-action for every object, and, for such $\beta$, the action extends to a $\bb{D}_{\nu,\epsilon}^{\operatorname{rat}}$-action, the specialization $\sf{AH}_1(\cal{R}(\beta))$ contains $L_\nu(1)$ with multiplicity one, and the filtration $\bm{f}_{\leq *}$ arises from the difference of its two gradings. This is proposed as a conjectural deformation motivated by the expected nonabelian Hodge-type correspondence; no resolution is stated.