Exponential complexity conjecture for local maxima of mixed p-spin Hamiltonians

Let $H_N$ be a mixed $p$-spin Hamiltonian on the sphere, let $\mathcal{N}(D)$ denote the number of its local maxima on $S_{N-1}$ whose normalized radial derivative lies in $D$, and let

$$I(x):=\frac{1}{2}\ln\left(\frac{\boldsymbol{\xi}”}{\boldsymbol{\xi}'}\right)-\frac{x^2}{4\boldsymbol{\xi}”}\frac{\boldsymbol{\xi}”-\boldsymbol{\xi}'}{\boldsymbol{\xi}”+\boldsymbol{\xi}'}+\Omega\left(\frac{x}{\sqrt{2\boldsymbol{\xi}”}}\right),$$

where $\boldsymbol{\xi}'=\xi'(1)$, $\boldsymbol{\xi}”=\xi”(1)$, and

$$\Omega(y):=\begin{cases}-\frac{1}{2}y\sqrt{y^2-2}+\ln\left(\frac{y+\sqrt{y^2-2}}{\sqrt{2}}\right)&\text{if }y\geq\sqrt{2},\\-\infty&\text{otherwise.}\end{cases}$$

Define $r_\infty:=2\sqrt{\boldsymbol{\xi}”}$ and $r_0:=\sup\{x\in\mathbb{R}:I(x)\geq0\}$. Exponential complexity conjecture. For all $r\in[r_\infty,r_0]$,

$$\lim_{\varepsilon\searrow0}\lim_{N\rightarrow\infty}\frac{1}{N}\ln\mathcal{N}([r-\varepsilon,r+\varepsilon])=I(r)$$

in probability. The theorem preceding this conjecture establishes matching first and second moments on the exponential scale and excludes local maxima outside the relevant radial-derivative interval; the conjecture asserts the corresponding quenched complexity formula for the actual count, rather than only its expectation.