Gross's newform conjecture for generic representations of split odd special orthogonal groups

Let $F$) be a finite extension of $\mathbb{Q}_p$, and let ${\rm SO}_{2n+1}(F)$ be the split odd special orthogonal group. Let $K_{n,m}$ and $J_{n,m}$ be Gross's families of open compact subgroups, with $K_{n,0}=J_{n,0}$ and $K_{n,m}$ a normal subgroup of $J_{n,m}$ of index $2$ for $m\geq 1$. Let $U_n(F)$ be the maximal unipotent subgroup, and fix the non-degenerate character $\psi_{U_n}$ defined from an unramified additive character $\psi$ of $F$. For an irreducible generic representation $\pi$ of ${\rm SO}_{2n+1}(F)$, let $\phi_\pi$ be its associated $L$-parameter and write

$$\epsilon(s,\phi_\pi,\psi)=\varepsilon_\pi q^{-c_\pi(s-\frac12)},$$

where $\varepsilon_\pi\in\{\pm1\}$, $c_\pi\geq 0$, and $q$ is the residue-field cardinality.

Gross's conjecture. The following assertions hold: $\pi^{K_{n,m}}=0$ for $0\leq m<c_\pi$ and $\dim_\mathbb{C}\pi^{K_{n,c_\pi}}=1$; the action of $J_{n,c_\pi}/K_{n,c_\pi}$ on $\pi^{K_{n,c_\pi}}$ is the scalar $\varepsilon_\pi$; and the natural pairing

$$\operatorname{Hom}_{K_{n,c_\pi}}(1,\pi)\times\operatorname{Hom}_{U_n(F)}(\pi,\psi_{U_n})\longrightarrow\mathbb{C}$$

is non-degenerate.

This is the newform conjecture for generic representations of ${\rm SO}_{2n+1}(F)$. The paper states that the existence part remains conditional because of unresolved issues in earlier proofs; the conjecture is therefore not established in the cited literature.