  We review and update on a few conjectures concerning matrix permanent that
are easily stated, understood, and accessible to general math audience. They
are: Soules permanent-on-top conjecture${}^\dagger$, Lieb permanent dominance
conjecture, Bapat and Sunder conjecture${}^\dagger$ on Hadamard product and
diagonal entries, Chollet conjecture on Hadamard product, Marcus conjecture on
permanent of permanents, and several other conjectures. Some of these
conjectures are recently settled; some are still open. We also raise a few new
questions for future study. (${}^\dagger$conjectures have been recently settled
negatively.)
