Conjecture on the exact recurrence orders for Stern's diatomic array sequences

Let $u_r(n)$ be the sequence associated with $A_r$, and let $\operatorname{mo}(r)$ denote the minimum order of a linear recurrence with constant coefficients satisfied by $u_r(n)$ for all sufficiently large $n$.

Exact recurrence-order conjecture. The values are

$$\operatorname{mo}(2)=2,\qquad \operatorname{mo}(6)=4,$$

and otherwise

$$\operatorname{mo}(2s)=2\left\lfloor\frac{s}{3}\right\rfloor+3,\qquad s\neq 1,3,$$ $$\operatorname{mo}(6s+1)=2s+1,\qquad s\geq 0,$$ $$\operatorname{mo}(6s+3)=2s+1,\qquad s\geq 0,$$ $$\operatorname{mo}(6s+5)=2s+2,\qquad s\geq 0.$$

This is intended to be best possible after accounting for the conjectured eigenvalue multiplicities and the apparent superfluity of the eigenvalue $1$ for even indices. It remains unproved in the source.