Conjecture on the sixth Hankel determinant for convolution powers of Narayana polynomials

Let $N$ be the determinant size, let $i$ and $j$ denote its row and column indices, and let $t$ be the parameter in the convolution powers of the Narayana polynomials. Write $N=3n$, $N=3n+1$, or $N=3n+2$ according to the residue class of $N$ modulo $3$. Define

$$r_n(t)=1+3t^3+6t^6+\cdots+\binom{n+1}{2}t^{3(n-1)}+\binom{n+2}{2}t^{3n}+\binom{n+1}{2}t^{3(n+1)}+\cdots+t^{6n}.$$

Conjecture 7.6. The sixth Hankel determinant satisfies

$$\Delta^{(6)}_{i+j}=\begin{cases} (-1)^n t^{9n(n-1)/2}[n+1]^2_{t^3}, & \text{if }N=3n,\\ (-1)^n t^{3n(3n-1)/2}[n+1]^2_{t^3}, & \text{if }N=3n+1,\\ (-1)^{n+1}3t^{3n(3n+1)/2}[3]_t r_n(t), & \text{if }N=3n+2. \end{cases}$$

Here $[m]_q=1+q+q^2+\cdots+q^{m-1}$ denotes the standard $q$-number. This conjecture gives an explicit residue-class formula for the sixth Hankel determinant associated with convolution powers of Narayana polynomials; its status is not resolved in the supplied source material.