Does the relaxation rate scale as $g^{2}$ in the thermodynamic limit?

For $H = H_{\mathrm{XXZ}} + g V$ with $V$ a generic local integrability-breaking perturbation not of the weak-integrability-breaking type (for that type the rate scales as $g^4$, Surace and Motrunich 2023), at infinite temperature, compute the exponent a in $\Gamma(g) \sim g^a$ for the decay rate of the slowest quasi-conserved local current, with $L \to \infty$ taken before $g \to 0$. The golden-rule expectation is $a = 2$. Answer: the value of a, with evidence that finite-size effects are controlled (they dominate when the golden-rule rate is below the inverse traversal time of the system, or for $g$ below the size-dependent chaos threshold).

An infinite-system computation of $\Gamma(g)$ (memory-matrix or recursion-method continued fraction) agreeing with real-time numerics on systems longer than the decay length at the smallest $g$ studied.