= Solution
Let $P_k$ be orthogonal projection onto the column space of $X_k$, and let $\mu$ be the true mean. Then
$$
\operatorname{RSS}_k=\|(I-P_k)Y\|^2
\sim\chi^2_{n-p_k}(\lambda_k),
\qquad
\lambda_k=\|(I-P_k)\mu\|^2.
$$
Up to a common constant,
$$
\operatorname{AIC}_k\sim\chi^2_{n-p_k}(\lambda_k)+2p_k,
\quad
\operatorname{BIC}_k\sim\chi^2_{n-p_k}(\lambda_k)+p_k\log n.
$$
The true model has $p_k=p^*$ and $\lambda_k=0$; every other candidate has $p_k>p^*$ and $\lambda_k\geq0$. Its excess expected AIC is $\lambda_k+p_k-p^*>0$, while its excess expected BIC is $\lambda_k+(\log n-1)(p_k-p^*)>0$ when $n>e$. Thus either minimum expected criterion selects the true model.
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