After discussion with Prof. Shi-Shuo Fu and Zhi-Cong Lin, we have proved Conjecture 1. The detaild proof will be contained in a joint paper by Fu, Lin and me. We will release a preprint in arXiv few days later.

Conjecture 2, and (3) and (4) in my comments are relatively easy, they have been proved by my graduate student Chen-Kai Ren.

After discussion with Prof. Shi-Shuo Fu and Zhi-Cong Lin, we have proved Conjecture 1. For the details, see the preprint Proof of five conjectures relating permanents to combinatorial sequences by Fu, Lin and me available from http://arXiv.org/abs/2109.11506.

Conjecture 2, and (3) and (4) in my comments are relatively easy, they have been proved by my graduate student Chen-Kai Ren.