The local ratios conjecture for real translations

Assume the Riemann hypothesis. Fix $k\geq 1$ and real numbers $\alpha_1,\ldots,\alpha_m,\beta_1,\ldots,\beta_m$ such that $\beta_\ell\neq0$ for every $\ell$ and $\alpha_i\neq\beta_j$ for every $i,j$. Define

$$E(\alpha,\beta)= \begin{cases} e^{-\alpha+\beta},&\beta<0,\\ 1,&\beta>0. \end{cases}$$

Local ratios conjecture. For these fixed parameters,

$$\frac{1}{T}\int_T^{2T}\prod_{\ell=1}^m\frac{\zeta\left(\frac12+\frac{\alpha_\ell}{\log T}+it\right)}{\zeta\left(\frac12+\frac{\beta_\ell}{\log T}+it\right)}\,dt \sim \frac{\det\left(\frac{E(\alpha_i,\beta_j)}{\alpha_i-\beta_j}\right)}{\det\left(\frac{1}{\alpha_i-\beta_j}\right)}.$$

This is a first-order local asymptotic for ratios of zeta values near the critical line and is motivated by random-matrix models; it remains conjectural.