Hard-edge universality conjecture for products of two coupled real Gaussian matrices

Consider the real counterpart of the joint probability density function in the paper's equation (matrixpdf), with the same parameters and scalings as in Theorem (hardlimits). Real hard-edge conjecture. Theorem (hardlimits) should still hold, with different limiting kernels having a Pfaffian structure. In particular, the critical scale of $\mu$ should again be $1/N$. This predicts a real-matrix analogue of the complex hard-edge phase transition, but the relevant Pfaffian limiting kernels and their convergence remain to be established.