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.