Kurokawa's Deep Riemann Hypothesis for automorphic Euler products

Let $\pi$ be an irreducible cuspidal automorphic representation of $\mathrm{GL}_n$ over $\mathbb Q$, with associated $L$-function

$$L(s,\pi)=\prod_p\prod_{j=1}^n(1-\alpha_{j,p}p^{-s})^{-1}.$$

Let $\nu(\pi)=m(\operatorname{sym}^2\pi)-m(\wedge^2\pi)$, where $m(\rho)$ is the multiplicity of the trivial representation in $\rho$, and let $m=\operatorname{ord}_{s=1/2}L(s,\pi)$. Assume that $L(s,\pi)$ is entire. Kurokawa's Deep Riemann Hypothesis. The limit

$$\lim_{x\to\infty}\left((\log x)^m\prod_{p\leq x}\prod_{j=1}^n\left(1-\alpha_{j,p}p^{-1/2}\right)^{-1}\right)$$

exists and is nonzero, and equals

$$\frac{\sqrt{2}^{\,\nu(\pi)}}{e^{m\gamma}m!}L^{(m)}\left(\frac12,\pi\right).$$

This conjecture concerns convergence of Euler products at the critical line and is used in the source as a framework for studying Chebyshev's bias for automorphic $L$-functions. The source gives no resolution status for the stated generality.