Extend logarithmic lower bounds beyond the present Matérn parameters

Extend the logarithmic lower bounds for standard Gaussian Process Thompson Sampling beyond the one-dimensional Matérn-$5/2$ setting studied in the paper, in order to determine how widely the Gaussian-overshoot mechanism applies.

Background

The paper proves matching $\Theta(N^{-5/2}\sqrt{\log N})$ worst-case expected simple-regret bounds for discrete and continuous GP-TS only for a fixed one-dimensional Matérn-$5/2$ kernel. The authors explicitly identify extending these lower bounds to other Matérn smoothness parameters and dimensions as unresolved, because doing so would clarify the generality of the identified competition between Gaussian extremes on suboptimal regions and sampled values near the optimum.