Determine whether non-quadratic strongly log-concave potentials are harder than Gaussians

Determine whether sampling from non-quadratic strongly log-concave potentials satisfying $I_d\preceq\nabla^2V\preceq\kappa I_d$ is strictly harder, in query complexity, than sampling from the centered Gaussian subclass.

Background

The paper’s lower bounds for centered Gaussian sampling also apply to strongly log-concave sampling with exact value-and-gradient queries, even when the minimizer is known. For Gaussian targets, the resulting lower bounds are nearly tight in several parameter regimes, whereas the cited general upper bounds for strongly log-concave sampling are substantially larger.

This motivates the unresolved comparison between the Gaussian subclass and genuinely non-quadratic strongly log-concave potentials: it is unknown whether non-quadratic structure creates additional sampling difficulty beyond the Gaussian case.