= Solution
Use the deterministic evidence-synthesis relation
$$
\pi_2=\pi_1+\lambda,
$$
with support
$$
\mathcal T=\{(\pi_1,\lambda):\pi_1\geq0,
\lambda\geq0, \pi_1+\lambda\leq1\}.
$$
A uniform prior on this triangle has density $2\mathbf1_{\mathcal T}$. Conditional on the prevalences, model the independent surveys by
$$
Y_{S1}\sim\operatorname{Binomial}(n_{S1},\pi_1),
\qquad
Y_{S2}\sim\operatorname{Binomial}(n_{S2},\pi_2),
$$
with the two counts <conditionally independent> and $\pi_2=\pi_1+\lambda$.
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