Computational hardness conjecture for the spiked Wishart hypothesis test

Let $\beta\geq -1$ and $\gamma>0$, and for $n\in\mathbb{N}$ set $N=\lceil n/\gamma\rceil$. Under $\mu_0$, draw $\mathbf{z}_1,\ldots,\mathbf{z}_N\sim N(0,I_n)$ independently. Under $\mu_1$, first draw $\mathbf{u}\sim\mathrm{U}(\{\pm1\}^n)$, then, conditional on $\mathbf{u}$, draw $\mathbf{z}_1,\ldots,\mathbf{z}_N\sim\mathsf{N}(0,I_n+(\beta/n)\mathbf{u}\mathbf{u}^{\top})$. Denote these two distributions collectively by $\mathrm{Wishart}(\beta,\gamma)$. A polynomial-time hypothesis test is a two-valued function of $(\mathbf{z}_1,\ldots,\mathbf{z}_N)$ evaluable in polynomial time in $n$, and it is asymptotically consistent if its probabilities of correctly identifying $\mu_0$ and $\mu_1$ both tend to $1$. Computational hardness conjecture for the spiked Wishart hypothesis test. If $\beta>-1$ and $\beta^2<\gamma$, there does not exist an asymptotically consistent sequence of polynomial-time hypothesis tests between the distributions of $\mathrm{Wishart}(\beta,\gamma)$. This conjecture posits an average-case computational barrier for distinguishing the null Wishart model from the planted rank-one spiked model in the stated parameter regime; the parser supplies no evidence that it has been resolved.