Novodvorsky compatibility conjecture for GSp(4) × GL(2) local L-factors

Let $F$ be a nonarchimedean local field of characteristic $0$, and let $\pi$ and $\sigma$ be generic irreducible smooth representations of $\operatorname{GSp}(4,F)$ and $\operatorname{GL}(2,F)$, respectively. Let $L(\pi \times \sigma,s)$ denote the Langlands–Shahidi local $L$-factor, and let $L^{\mathrm{Nov}}(\pi \times \sigma,s)$ denote Novodvorsky's local $L$-factor.

Novodvorsky compatibility conjecture. We have

$$L(\pi \times \sigma,s)=L^{\mathrm{Nov}}(\pi \times \sigma,s)$$

for all generic $\pi$ and $\sigma$.

This compatibility is known in a substantial range of cases, including when $\sigma$ is non-supercuspidal, but many cases remain open.