Continuum conjecture for unitarily non-equivalent MHDRs of bivariate polynomials

A multivariate Hermitian determinantal representation (MHDR) of a polynomial is a determinantal representation using Hermitian coefficient matrices, and two MHDRs are unitarily equivalent when they are related by unitary conjugation. For a bivariate polynomial of degree $d$, the conjecture is:

Continuum conjecture. There is a continuum of unitarily non-equivalent MHDRs for a bivariate polynomial of degree $d$.

The claim is motivated by constructions for cubic and quartic bivariate polynomials, where varying phase parameters produces distinct equivalence classes. The source presents these examples as experimental evidence; the general assertion for degree $d$ remains unproved here.