Polynomiality conjecture for Hankel determinants of generalized Catalan powers

Let $G(x,r)$ be the generalized generating function used in the paper, and write $H_n(G(x,r))$ for the Hankel determinant of its coefficients. For a positive integer $t$, consider first $r=2t+1$. Generalized Catalan polynomiality conjecture. If $r=2t+1$ is odd and $1\leq j\leq\left\lfloor\frac{r+3}{4}\right\rfloor$, then

$$H_{rn+j}(G(x,r)),\quad H_{rn+t+j}(G(x,r)),\quad H_{rn+t+2-j}(G(x,r)),\quad H_{rn+r+1-j}(G(x,r))$$

are all polynomials in $n$ of degree $(2j-1)(t+1-j)$. If $r=2t$ is even and $1\leq j\leq t$, then

$$(-1)^{tn}H_{rn+j}(G(x,r))$$

and

$$(-1)^{tn}H_{rn+r+1-j}(G(x,r))$$

are both polynomials in $n$ of degree $(j-1)(r+1-2j)$. The conjecture extends the paper's polynomiality predictions to the generalized family $G$; the supplied context gives no resolution, so its status remains open.