Failure or validity of Gaussian lower tails in dimensions three and higher

Determine whether the Gaussian lower-tail estimate established for coordinate-periodic, $2\times2$ uniformly elliptic systems driven by fractional Brownian motion in Young’s regime fails in dimension three and higher.

Background

Theorem D proves that every coordinate-periodic, $2\times2$ uniformly elliptic system has a Gaussian lower tail in Young’s regime $H\in(1/2,1)$. This result contrasts with the paper’s radially periodic counterexamples, which have a lighter $(2H+1)$-Weibull upper tail.

The authors explicitly indicate that the two-dimensional periodic result may not extend to higher dimensions. Whether an analogous Gaussian lower-tail theorem remains valid, or instead fails, for periodic uniformly elliptic systems in dimension three or higher is left unresolved.