= Solution
For parameters $(\rho_G,\pi_1,\lambda)$ on $0<\rho_G<1$ and $(\pi_1,\lambda)\in\mathcal T$, conditional independence gives
$$
\begin{aligned}
\pi(\rho_G,\pi_1,\lambda\mid y_G,y_{S1},y_{S2})
\propto{}&
(\lambda\rho_G)^{y_G}(1-\lambda\rho_G)^{N-y_G}\\
&\times\pi_1^{y_{S1}}(1-\pi_1)^{n_{S1}-y_{S1}}\\
&\times(\pi_1+\lambda)^{y_{S2}}
(1-\pi_1-\lambda)^{n_{S2}-y_{S2}}\\
&\times\rho_G^{a-1}(1-\rho_G)^{b-1}
\mathbf1_{\mathcal T}(\pi_1,\lambda).
\end{aligned}
$$
This is the joint posterior up to its normalizing constant.
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