Let $X \in \mathbb{R}^{m \times n}$ and $Y \in \mathbb{R}^{n \times k} $ be two independent Gaussian random matrices, i.e., with entries independently sampled from $\mathcal{N}(0,1)$ (a normal distribution with zero mean and unit variance). We define the positive orthant probability $$p_+(m,n,k) \triangleq P(\forall i,j: (XY)_{ij} >0).$$ Question: How does $\log_2 p_+(m,n,k)$ behave asymptotically, to leading order, in the limit $ n \rightarrow \infty$ and $\alpha m = \beta k = n $, for some positive constants $\alpha$ and $\beta$?
Note that if $n\rightarrow \infty$ but $m$ and $k$ remain fixed, then from the central limit theorem $\frac{1}{\sqrt{n}}XY$ becomes a Gaussian matrix with independent entries, and then we get $\log_2 p_+(m,n,k) = -mk$. However, I wish to know if how this asymptotic behavior persists when $m$ and $k$ are of similar magnitude as $n$. Specifically, what is the largest positive constant $r>1$ for which we have
$$\lim_{\underset{\alpha m = \beta k = n}{n\rightarrow\infty}} \frac{1}{n^{r}} \log_2 p_+(m,n,k) < 0 \,.$$
I guess the answer should depend on $\alpha$ and $\beta$. Note I relaxed the conditions in this question from the previous version, since I didn't get an answer so far.
Thanks in advance!