Conjecture on shifted periodicity of Hankel determinants for Motzkin convolution powers

Let $F(x,r)=M(x)^r$, where $M(x)$ is the generating function of the Motzkin numbers, and let $H_n(F(x,r))$ be its Hankel determinants. The conjecture asserts that, for $r\equiv 0\pmod 3$, the shifted period of $H_n(F(x,r))$ is $r$, whereas, for $r\equiv 1$ or $2\pmod 3$, its shifted period is $3r$. This summarizes the periodic patterns predicted by the preceding formulas for the Hankel determinants of Motzkin convolution powers.