The newform zeta-integral conjecture for unramified U(2,1)

Let $E/F$ be an unramified quadratic extension of non-archimedean local fields of characteristic zero and odd residual characteristic. Fix an additive character $\psi_E$ of $E$ with conductor $\mathfrak{o}_E$. Let $\pi$ be an irreducible generic representation of $\mathrm{U}(2,1)(E/F)$, and let $N_\pi$ denote its conductor. Write $W_v$ for the Whittaker function associated to a newform $v$, and let $\Phi_{N_\pi}$ be the characteristic function of

$$\mathfrak{p}_F^{N_\pi}\oplus\mathfrak{o}_F.$$

Newform zeta-integral conjecture. There exists a newform $v$ for $\pi$ such that

$$Z(s,W_v,\Phi_{N_\pi})=L(s,\pi).$$

This conjecture predicts that the local $L$-factor of an irreducible generic representation of unramified $\mathrm{U}(2,1)$ is represented by the zeta integral attached to a suitable newform and the canonical test function. It is the analogue for $\mathrm{U}(2,1)$ of the newform results for general linear groups; the supplied text does not state whether it has been proved or disproved.