Conjecture on Hankel determinants of Motzkin convolution powers when the exponent is divisible by 3

Let $M(x)=\sum_{n\geq 0}M_nx^n$ be the generating function of the Motzkin numbers, let $F(x,r)=M(x)^r$, and write $H_n(F(x,r))$ for the Hankel determinant of the coefficients of $F(x,r)$. For $r\equiv 0\pmod 3$, the conjecture asserts

$$H_{rn}(F(x,r))=H_{rn+1}(F(x,r))=(-1)^n(n+1)^{r-1}.$$ $$H_{rn+2}(F(x,r))+H_{rn-1}(F(x,r))=3^2\alpha(n+1)^{r-1},$$

where $|\alpha|=\frac{1}{18}r(r-3)$. These formulas extend the explicit patterns observed for small convolution powers and predict a structured form for the Hankel determinants when the exponent is a multiple of $3$.