The normalized period conjecture in categorical local Langlands

Let $G$) be a connected quasi-split reductive group over $F$ with a fixed Whittaker datum. Suppose there is a categorical equivalence

$$\mathbb{L}_G \colon {\mathcal{D}}^{\operatorname{oc}}({\operatorname{Bun}}_G, {\overline{\mathbb{Q}}_\ell}) \cong {\operatorname{IndCoh}}({\operatorname{Par}}_{{{}^LG}}).$$

For a dual pair $(G,X)\leftrightarrow({{}^LG},\widehat{X})$, the objects ${\mathcal{P}}_X^{\operatorname{norm}}$ and ${\mathcal{L}}_{\widehat{X}}^{\operatorname{norm}}$ are the normalized period and normalized $L$-object, respectively. The normalized period conjecture. For every conjecturally defined dual pair $(G, X) \leftrightarrow ({{}^LG}, \widehat{X})$, one has

$$\mathbb{L}_G({\mathcal{P}}_X^{\operatorname{norm}}) \cong {\mathcal{L}}_{\widehat{X}}^{\operatorname{norm}}.$$

This is the categorical translation of the normalized period conjecture in the setting of local Langlands; it is conditional on the existence of the stated categorical equivalence and on the definition of the relevant dual pairs.