The fullness conjecture for cyclotomic web functors

Let $\Bbbk$ be a field, let $\mathpzc{Web}_{\mathbf u}$ be the cyclotomic web category, let $W(\Lambda)$ be the associated $W$-algebra, and let $\widehat\Phi$ be the functor from $\mathpzc{Web}_{\mathbf u}$ to $W(\Lambda)\text{-mod}$. Let $\mathpzc{Web}_{\mathbf u,N}$ be the quotient category defined from the finite-$N$ affine web category. The fullness conjecture. Suppose that $\Bbbk=\mathbb C$. The functor

$$\widehat\Phi:\mathpzc{Web}_{\mathbf u}\longrightarrow W(\Lambda)\text{-mod}$$

is full. Moreover, it induces a fully faithful functor from $\mathpzc{Web}_{\mathbf u,N}$ to $W(\Lambda)\text{-mod}$. The statement extends the preceding asymptotic faithfulness theorem to fullness and to the finite-$N$ quotient; the supplied text gives no resolution status.