Weak form of the Main Conjecture for circuit-array rows

Let $C^X_r(j)$ and $C^M_r(j)$ denote the entries in rows $r$ of the corresponding circuit arrays, and let $Num$ and $Den$ denote their numerators and denominators. A sequence satisfies a recursion when it has an associated annihilator polynomial in the shift operator. For each row $r \in \{2(c-1),2(c-1)-1\}$, consider the sequences indexed by $j \ge c$. Weak form of the Main Conjecture. Each of the sequences $Num[C^X_r(j)]$, $Den[C^X_r(j)]$, $Num[C^M_r(j)]$, and $Den[C^M_r(j)]$ satisfies a recursion whose corresponding annihilator is a product, possibly with repetition, of linear factors of the form $X-9^k$ with $k \ge 0$. This is the weaker formulation of the paper's proposed description of the recursions governing the circuit-array rows; the source gives no general proof or resolution, although the stronger form would imply it.