Invertibility conjecture for Gram matrices of monomial-potential orthogonal polynomials

Let $d$ be a positive integer and $a\geq0$, and consider the hermitian inner product with density

$$\rho_a(z)=\exp\left(-a|z|^2+\frac{i}{d}(z^d+\bar z^d)\right).$$

For each nonnegative integer $N$, let $G_N=((z^n,z^m))_{n,m=0}^N$ be its Gram matrix on the monomial basis. Gram-matrix invertibility conjecture. The matrices $G_N$ are invertible for every $N=0,1,2,\dots$ and every $a\geq0$. This would provide a criterion ensuring the existence of the associated orthogonal polynomials for all finite degrees; the source gives no resolution.