= Solution
Let $\eta_i=\alpha+\beta\log(x_i+10)+\gamma x_i$. The <Poisson-lognormal random-effect model> assigns each plate a random intensity $\mu_{ij}=e^{\eta_i+\lambda_{ij}}$. Conditional on that intensity its <Poisson distribution> has mean and variance both $\mu_{ij}$, but marginalizing over the <normal distribution> of the <random effect> gives
$$
m_i=\mathbb E[Y_{ij}\mid\alpha,\beta,\gamma,\tau]
=e^{\eta_i+\tau^2/2}.
$$
The <law of total variance> yields
$$
\operatorname{Var}(Y_{ij})=\mathbb E[\mu_{ij}]+\operatorname{Var}(\mu_{ij})
=\boxed{m_i+m_i^2(e^{\tau^2}-1)}.
$$
This exceeds the mean whenever $\tau>0$. The independent plate effects supply additional heterogeneity without inducing dependence between plates conditional on the global parameters. Also $e^{\eta_i}$ is the median random intensity, not its marginal mean; the factor $e^{\tau^2/2}$ matters for population predictions.
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