Bessenrodt and Ono's work on additive and multiplicative properties of the partition function and DeSalvo and Pak's paper on the log-concavity of the partition function have generated many beautiful theorems and conjectures. In January 2020, the first author gave a lecture at the MPIM in Bonn on a conjecture of Chern--Fu--Tang, and presented an extension (joint work with Neuhauser) involving polynomials. Partial results have been announced. Bringmann, Kane, Rolen and Tripp provided complete proof of the Chern--Fu--Tang conjecture, following advice from Ono to utilize a recently provided exact formula for the fractional partition functions. They also proved a large proportion of Heim--Neuhauser's conjecture, which is the polynomization of Chern--Fu--Tang's conjecture. We prove several cases, not covered by Bringmann et.\ al. Finally, we lay out a general approach for proving the conjecture.