An update on a few permanent conjectures

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.)