= Solution
A <proper gamma approximation to a Poisson Jeffreys prior> uses shape $1/2$ and a small positive rate $\varepsilon$:
$$
\boxed{\lambda_i\sim\operatorname{Gamma}(1/2,\varepsilon),\qquad\varepsilon>0\ \text{small}.}
$$
Its density is proportional to $\lambda_i^{-1/2}e^{-\varepsilon\lambda_i}$, so where $\varepsilon\lambda_i$ is small its kernel approximates the <Jeffreys prior>. After observing the data, <Poisson-gamma conjugacy> gives $\operatorname{Gamma}(y_i+1/2,E_i+\varepsilon)$, which converges to the preceding <posterior distribution> as $\varepsilon\downarrow0$. This is an approximation of an improper kernel on the relevant parameter range, not a weak limit to a normalized Jeffreys probability distribution. The shape must be $1/2$; a small-shape gamma prior would approximate a different power of $\lambda_i$.
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