Asymptotic minimaxity of the $2k\log(n/k)$ penalized model-selection procedure

Let model selection mean adaptive selection of nonzero means, and let the underlying estimand be $bc$, belonging to one of the parameter spaces and considered under one of the losses covered by Theorem. Let the procedure referred to as

be the corresponding $2k\log(n/k)$ penalized procedure. **Asymptotic minimaxity conjecture.** The procedure

is asymptotically minimax simultaneously over the full range of parameter spaces and losses covered by that theorem. This conjecture concerns whether the variable-complexity penalty suggested by the paper achieves optimal risk across all of the theorem's model-selection settings; the supplied text gives no resolution.