Extension of the lower bounds to local oracles

The functions $p_1$ and $p_2$ are constructed in the preceding results, and Theorem~ and Corollary~ give lower bounds for deterministic second-order optimization under the stated assumptions. A local oracle is the oracle model in which the information returned at a query depends only on the function's local behavior.

Extension conjecture. Theorem~ and Corollary~ remain valid (in substance) even under local oracles.

The proposed extension would show that the lower bounds are not specific to second-order oracles. The authors explain that flattening $p_1$ near $0$ suggests a route to a local-oracle construction, but they leave the corresponding analysis open because the resulting function is piecewise polynomial in norms rather than piecewise polynomial.