In van Dijk's notes of Harish Chandra's lectures on harmonic analysis, several conjectures are mentioned throughout, such as in Part 1, section 4 of van Dijk's notes
Conjecture I : Let $\omega$ be an equivalence class of a square integrable representations of a $p$-adic reductive group $G$, and $d(\omega)$ it's formal degree. Let $\mathcal{E}(G)$ denote the equivalence classes of irreducible unitary representations of $G$, and $\mathcal{E}^{0}(G)$ the subset consisting of supercuspidals, then $\mathrm{inf}_{\mathcal{E}^{0}(G)} d(\omega) >0$
and (part 4, section 1)
Conjecture III: Let $(\tau,V)$ be a finite-dimensional unitary representation on a Hilbert space $V$ of a "good" maximal compact $K$ (in the sense of Bruhat-Tits). Let $\mathcal{H}(\tau)$ denote the $(\tau,K)$ Hecke algebra, i.e. the algebra of compactly supported functions $\beta: G \to \mathbb{C}$ satisfying $\beta(k_1xk_2)=\tau(k_1)\beta(x)\tau(k_2)$. For a parabolic $P=MN$, let $\mu_P: \beta \mapsto \beta^{(P)}$ denote the constant term along $P$. The image $\mu_P(\mathcal{H}(\tau))$ lies inside $\mathcal{H}(\tau_M)$, where we regard $\tau_M$ as a representation of $K\cap M$ by first projecting onto the subspace of vectors $v\in V$ such that $\tau(n)v=v$ and then letting $m$ act through $\tau$. Then $\mathcal{H}(\tau_M)$ is a finite right module over $\mu_P(\mathcal{H}(\tau))$.
As Harish Chandra mentions, this is proved for $\tau = \mathrm{Id}$ and $P$ the minimal parabolic of a simply connected reductive group over char 0 field, by Satake in his "Satake - isomorphism" paper.
These conjectures are used pretty often throughout the rest of the notes, so I wonder whether they are well understood by now.
