= Solution
Let $U_i=\{j:(i,j)\notin O\}$ and let $x_{i,O}$ denote the observed entries in row $i$. Conditional on the parameters, different rows of the missing design are independent, while the missing entries within one row are coupled by its Gaussian response. For $z\in\{0,1\}^{U_i}$,
$$
\Pr(X_{i,U_i}=z\mid Y,X_O,\beta,\pi)
\propto
\exp\left[-\frac{(Y_i-x_i(z)^T\beta)^2}{2\sigma^2}\right]
\prod_{j\in U_i}\pi_j^{z_j}(1-\pi_j)^{1-z_j}.
$$
Normalizing this expression over the $2^{|U_i|}$ configurations and multiplying over rows gives the full conditional distribution of $X_U$.
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