Canonical Higgsing conjecture for generic quiver $qq$-characters

Let $\Gamma$ be a generic quiver and let $w=(w_i)_{i\in\Gamma_0}\in\mathbb{Z}_{\ge 0}^{\operatorname{rk}\Gamma}$ be a generic weight dimension vector. Let $\underline{x}=(x_{i,\alpha})$ denote the weight parameters. Canonical Higgsing conjecture. There exists a canonical specialization of $\underline{x}$ such that the $qq$-character is reduced to the $qq$-character of the irreducible highest-weight module parametrized by $w$. This conjecture concerns generic modules and generic quivers; the paper relates it to tensor product modules of Fock modules for quantum toroidal algebras associated with the $\widehat{A}_0$ and $\widehat{A}_{r-1}$ quivers, while the general assertion remains unproved.