Charged quantum-system chaos bound with an effective inverse temperature

Let $H$ be the Hamiltonian, $Q$ a conserved charge, and define

$$y^4=\frac{e^{-\beta H}}{Z},\qquad z^4=e^{\beta\mu Q}.$$

For operators that can create states with arbitrarily large values of $Q$, consider the regularized out-of-time-order correlator

$$F(t)=\operatorname{tr}\left[y z W(t)y z V(0)y z W(t)y z V(0)\right].$$

Assume that $F$ is well behaved for $t\geq0$ and that the Hilbert space satisfies the source's condition for a fixed constant $\mu_c$. Charged chaos-bound conjecture. Writing $f(t')=F(t_d+t')/F_*$, where $F_*$ is a suitable time-independent factor, one has

$$\frac{1}{1-f(t')}\left|\frac{df(t')}{dt'}\right|\leq\frac{2\pi}{\beta_{\mathrm{eff}}}+\mathcal{O}\left(e^{-\frac{4\pi}{\beta_{\mathrm{eff}}}t'}\right), \qquad \beta_{\mathrm{eff}}=\beta\left(1-\left|\frac{\mu}{\mu_c}\right|\right).$$

The authors expect this formula near the scrambling time, $0\ll t'<t_*-t_d$, with all other insertion coordinates fixed and for $|\mu/\mu_c|\ll1$, at temperatures much higher than the scale of any compact dimension when present. Its validity beyond this regime, and the precise content of the referenced Hilbert-space condition, remain open.