Polynomiality conjecture for even Catalan convolution Hankel determinants

Let $C(x)=\sum_{n\geq 0}C_nx^n$ be the Catalan generating function, let $F(x,r)=C(x)^r$, and write $H_n(F(x,r))$ for the $n\times n$ Hankel determinant of the coefficients of $F(x,r)$. For a positive integer $t$, set $r=2t$. Even polynomiality conjecture. For $1\leq j\leq t$, both

$$(-1)^{n\binom{t}{2}}H_{tn+j}(F(x,r))$$

and

$$(-1)^{n\binom{t}{2}}H_{tn+t-j+1}(F(x,r))$$

are polynomials in $n$ of degree $(2j-1)(t-j)$. This conjecture predicts the polynomial structure underlying the observed periodic Hankel determinant patterns; computations for convolution powers up to $r=31$ support it, but the paper does not prove it.