Conjecture on archimedean Rankin–Selberg zeta integrals for GL_n

Let $\pi^{(n+1)}$ and $\pi^{(n)}$ be cohomological irreducible cuspidal automorphic representations of $\operatorname{GL}_{n+1}(\mathsf{F}_{\mathbf{A}})$ and $\operatorname{GL}_n(\mathsf{F}_{\mathbf{A}})$, respectively, satisfying the paper's automorphic assumptions and the interlace condition. Let $N$ be either $n+1$ or $n$. For each archimedean place $v\in\Sigma_{\mathsf{F},\infty}$ and each signature $\varepsilon^{(N)}_v\in\{\pm1\}$ for which the relevant $(\mathfrak{g}_{N,v},K^{(N)}_v)$-cohomology group is nonzero, choose a nontrivial cohomology class $[\pi^{(N)}_v]^{\varepsilon^{(N)}_v}$.

Conjecture on archimedean zeta integrals. There exists a number field $E(\varepsilon^{(n+1)}_v,\varepsilon^{(n)}_v)$ such that, for every $m\in\mathrm{Crit}(\pi^{(n+1)},\pi^{(n)})$, the archimedean local zeta integral at $s=\frac12+m$ satisfies

$$\widetilde{\mathcal I}_v\left(s,[\pi^{(n+1)}_v]^{\varepsilon^{(n+1)}_v},[\pi^{(n)}_v]^{\varepsilon^{(n)}_v},\varphi_v\right)\bigg|_{s=\frac12+m}\sim_{E(\varepsilon^{(n+1)}_v,\varepsilon^{(n)}_v)}L_v\left(\frac12+m,\pi^{(n+1)}\times\pi^{(n)}\times\varphi\right).$$

At a real place, this is asserted when the signatures satisfy $\varphi_v(-1)\varepsilon^{(n+1)}_v\varepsilon^{(n)}_v=(-1)^{m+n+1}$. The conjecture concerns the algebraicity of critical Rankin–Selberg values through the difficult archimedean local zeta integrals; it is stated to hold in several cases, but no general resolution is supplied here.