Conjecture on Hankel determinants of Motzkin convolution powers when the exponent is not 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 $r\equiv 1$ or $2\pmod 3$, the conjecture asserts

$$H_{3rn}(F(x,r))=H_{3rn+1}(F(x,r))=H_{3rn+r}(F(x,r))=H_{3rn+r+1}(F(x,r))=\alpha.$$ $$H_{3rn+2r}(F(x,r))=H_{3rn+2r+1}(F(x,r))=0.$$ $$H_{3rn+2r-1}(F(x,r))+H_{3rn+2r+2}(F(x,r))=\beta\left((2r)(n+1)\right)^{r-2}.$$ $$H_{3rn+2}(F(x,r))+H_{3rn-1}(F(x,r))=\frac12\gamma r(r-3),$$

where $|\alpha|=|\beta|=|\gamma|=1$. The conjecture describes the observed block structure, vanishing determinants, and paired sums for convolution powers whose exponent is congruent to $1$ or $2$ modulo $3$.