The modified local Langlands family conjecture

Let $G={\rm GL}_n(F)$, let $X_{\check G}$ be the space of L-parameters $(\varphi,N)$ for the principal block, and let $\widetilde{\mathcal{V}}_G$ be the family of smooth $G$-representations over $X_{\check G}$ constructed from the generic points and the modified local Langlands correspondence. The modified local Langlands family conjecture. For every point $x=(\varphi,N)\in X_{\check G}$,

$$(\widetilde{\mathcal{V}}_G\otimes k(x))^\vee\cong {\rm LL}^{\rm mod}((\varphi,N)^\vee).$$

This asserts that the family interpolates the modified local Langlands correspondence at every, including nongeneric, parameter. The source presents it as conjectural and does not prove the claimed fiberwise identification.