Optimality of the exponent one-half in convergence rates for dynamic random Hamilton–Jacobi equations

Consider the setting of Theorems --, with the convergence rates established there. Optimality conjecture. The convergence rates obtained therein are optimal up to slowly varying factors. In particular, the optimal convergence rate is of exponent $1/2$. The conjecture asserts that the exponent one-half in the paper's quantitative convergence results cannot be improved, apart from slowly varying factors.