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

Let $G_n={\rm SO}_{2n+1}$ be the split odd special orthogonal group of rank $n\geq 1$ over $F$, let $U_n\subset G_n$ be its upper triangular maximal unipotent subgroup, and let $(\pi,\mathcal V_\pi)$ be an irreducible smooth generic representation of $G_n(F)$. Fix an additive character $\psi$ of $F$ with $\ker(\psi)=\mathfrak{o}$, a corresponding non-degenerate character $\psi_{U_n}$, and a nonzero Whittaker functional $\Lambda_{\pi,\psi}\in {\rm Hom}_{U_n(F)}(\pi,\psi_{U_n})$. Let $\phi_\pi$ be the $L$-parameter attached to $\pi$, and write the epsilon factor associated to its composition with ${}^LG_n^0(\mathbb C)={\rm Sp}_{2n}(\mathbb C)\hookrightarrow {\rm GL}_{2n}(\mathbb C)$ as

$$\epsilon(s,\phi_\pi,\psi)=\varepsilon_\pi q^{-a_\pi(s-\genfrac{}{}{}{}{1}{2})},$$

where $\varepsilon_\pi\in\{\pm1\}$ and $a_\pi\geq 0$. For the subgroups $K_{n,m}\subset G_n(F)$ and $J_{n,m}$ used in the newform theory, Tsai's newform conjecture. For every $m\geq 0$,

$$\dim_\mathbb C\mathcal V_\pi^{K_{n,m}}=\genfrac{(}{)}{0pt}{}{n+\left\lfloor\genfrac{}{}{}{}{m-a_\pi}{2}\right\rfloor}{n}+\genfrac{(}{)}{0pt}{}{n+\left\lfloor\genfrac{}{}{}{}{m-a_\pi+1}{2}\right\rfloor-1}{n}.$$

In particular, $\mathcal V_\pi^{K_{n,m}}=0$ if $0\leq m<a_\pi$, and $\mathcal V_\pi^{K_{n,a_\pi}}$ is one-dimensional; the action of $J_{n,a_\pi}/K_{n,a_\pi}$ on $\mathcal V_\pi^{K_{n,a_\pi}}$ gives $\varepsilon_\pi$; and $\Lambda_{\pi,\psi}$ is nontrivial on $\mathcal V_\pi^{K_{n,a_\pi}}$. This conjecture predicts the dimensions and arithmetic character of newform and oldform spaces, as well as the nonvanishing of the Whittaker functional on the newform space; its status is not resolved in the supplied source context.