The quenched complexity formula for TAP critical points

Let $F_{\mathrm{TAP}}$ be the TAP free energy, and for $f\in\mathbb{R}$ and $\varepsilon>0$ let $\mathcal{N}_\varepsilon(f)$ denote the number of TAP critical points at free-energy level $f$. Define

$$\widetilde{\Lambda}(\theta)=\theta\inf_{\zeta:\,\zeta([0,\sup(\operatorname{supp}\zeta)))=\theta}\mathcal{P}\mathrm{arisi}(\zeta).$$

The quenched complexity formula. There exists an open subset $D\subset\mathbb{R}$ such that, for every $f\in D$,

$$\lim_{\varepsilon\to 0}\lim_{N\to\infty}\frac{1}{N}\mathbb{E}\bigl[\log\mathcal{N}_\varepsilon(f)\bigr]=-\widetilde{\Lambda}^*(f)=\inf_\theta\bigl(\widetilde{\Lambda}(\theta)-\theta f\bigr).$$

This formula is proposed for the quenched complexity, and the conditional annealed lower bound supplies evidence for it in the mixed case. The source notes that the relevant Parisi optimizer has a jump at the top of its support in certain regimes, but does not establish the formula in general.