= Solution
For each $r$, conditioning on all the other coordinates gives no greater <conditional entropy> than conditioning only on the preceding ones. Summing and using the <chain rule for conditional entropy>[chain rule for entropy] gives directly
$$
\boxed{\sum_rH(X_r\mid X_{-r})\leq\sum_rH(X_r\mid X_1,\ldots,X_{r-1})=H_n}.
$$
Equivalently, the left side is $nH_n-\sum_rH(X_{-r})$, so the same conclusion follows from <Han's entropy inequality>. The direct proof also covers $n=1$, interpreting conditioning on an empty vector as no conditioning.
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