5 problems

Real-eigenvalue count conjecture. With high probability, $M_n$ has $\Theta(\sqrt n)$ real eigenvalues.

Spectral fingerprint conjecture. Almost all $\pm1$ matrices are determined by their spectrum.

Gaussian-integral-spectrum conjecture. The probability that all eigenvalues of $M_n$ are Gaussian integers is

Integral-spectrum conjecture. The probability that all eigenvalues of $M_n^{\mathrm{sym}}$ are integers is

Symmetric spectrum simplicity conjecture.