On a tamely ramified local relative Langlands conjecture via categorical representations

Let $G$ be a complex reductive group. For a smooth affine spherical $G$-variety $X$, assume that the unramified relative local Langlands conjecture of Ben-Zvi-Sakellaridis-Venkatesh for $X$ holds, the loop space $LX$ is an $L^{+}G$–placid ind–scheme, and there exists a dimension theory for $LX$, we give a spectral description of a full subcategory of Iwahori equivariant D-modules on $LX$ in terms of the relative Langlands dual of $X$, confirming a slight variant of the tamely ramified local relative Langlands conjecture proposed by Devalapurkar.