The test function conjecture for Shimura varieties

Let $G$ be the reductive group in the Shimura datum, let $K_p\subset G(\mathbb Q_p)$ be the prescribed level subgroup, and let $K_{p^r}\subset G(\mathbb Q_{p^r})$ be its base change. Let $d={\rm \dim}(Sh_{K_p})$, and let $Z_{V^{E_{j0}}_{-\mu,j}}$ be the Bernstein-center distribution associated to the indicated representation of the Langlands $L$-group. Let $1_{K_{p^r}}$ denote the characteristic function of $K_{p^r}$. Test function conjecture. The test function $\phi_r$ in the semisimple Lefschetz formula may be taken to be

$$\phi_r=p^{rd/2}\bigl(Z_{V^{E_{j0}}_{-\mu,j}}*1_{K_{p^r}}\bigr).$$

In particular, it lies in the center $\mathcal Z(G(\mathbb Q_{p^r}),K_{p^r})$ of the Hecke algebra, and the test functions vary compatibly with changes in $K_p$. This conjecture was formulated jointly with R. Kottwitz; the source notes that some unconditional versions have been proved.