Hardness conjecture for the spiked Wishart model

Let $\beta \in [-1,\infty)$ and $\gamma>0$. For each $n\in\mathbb N$, set $N=\lceil n/\gamma\rceil$, and let $\mathbb P$ and $\mathbb Q$ be the spiked Wishart distributions defined by the null model $\mathbb Q$ with independent samples $\bm y_i\sim\mathcal N(\bm 0,\bm I_n)$ and the planted model $\mathbb P$ obtained by sampling $\bm x\sim\operatorname{Unif}(\{\pm1\}^n)$ and then independently sampling $\bm y_i\sim\mathcal N(\bm 0,\bm I_n+(\beta/n)\bm x\bm x^{\top})$. Write this sequence of distribution pairs as $\mathsf{Wishart}(\beta,\gamma)$. Hardness of the Wishart model. If $\beta>-1$ and $\beta^2<\gamma$, there is no polynomial-time hypothesis test between the distributions of $\mathsf{Wishart}(\beta,\gamma)$: no randomized algorithm $f:(\mathbb R^n)^N\to\{\mathtt p,\mathtt q\}$ running in polynomial time in $n$ satisfies

$$\lim_{n\to\infty}\mathbb P[f(\bm y_1,\dots,\bm y_N)=\mathtt p]=\lim_{n\to\infty}\mathbb Q[f(\bm y_1,\dots,\bm y_N)=\mathtt q]=1.$$

This conjecture is the key computational-hardness assumption used to obtain evidence that the constant $1$ in spectral conditions for general-purpose Ising-model sampling and free-energy approximation is optimal; its status is not resolved in the source.