The local Plancherel density conjecture for the basic function

Let $F$ be a split non-Archimedean local field with ring of integers $\mathfrak o$, and let $X$ be an affine spherical variety with a good model over $\mathfrak o$, satisfying the injectivity assumption $\check A_X\to\check A$. Let $\Phi_0$ be its basic function, and let $\check A_X^1/W_X$ denote the unramified parameter space. Local Plancherel density conjecture. There is a representation

$$\rho_X:{^LG_X}\to\operatorname{GL}(V_X)$$

such that, with $L_X(\phi)=L(\phi,\rho_X,0)$, the Plancherel decomposition is

$$\lVert\Phi_0\rVert^2=\int_{\check A_X^1/W_X}\frac{L_X(\phi)}{L(\phi,\check{\mathfrak g}_X,1)}\,\mu_{\check G_X}(\phi).$$

This predicts that the basic function’s Plancherel density is governed by a special value of the local $L$-function attached to $\rho_X$; the source presents it as an important open problem, especially beyond the unramified setting.