Strong form of the Main Conjecture for circuit-array rows

Let $C^X_{r,j}$ denote the entries of the $X$-circuit array. For $c=1,r=0$ and for each $c \ge 2$ with corresponding $r \in \{2(c-1),2(c-1)-1\}$, suppose there is a sequence of polynomials $p_r(j,X)$ for $j \ge c$. Define the recursive array $e$ by the rules in the claim, with all otherwise undefined entries equal to zero. Strong form of the Main Conjecture. If $r=2(c-1)-1$, then there is a constant $K$ depending at most on $c$ such that

$$C^X_{r,j}=K\frac{p_{r-3}(j-1)p_{r-1}(j)}{p_{r-2}(j)p_r(j)}.$$

If $r=2(c-1)$, then there is a constant $K$ such that

$$C^X_{r,j}=K\frac{p_{r-4}(j-1)p_r(j)}{p_{r-1}(j)^2}.$$

Moreover, for $c \ge 1$ and $r \in \{2(c-1),2(c-1)+1\}$, $\deg(p_r)=c$; the degree-$0$ coefficient sequence has annihilator

$$\prod_{k=0}^c(X-9^k)^{e_{2r,0,k}},$$

and for $j>0$ the degree-$j$ coefficient sequence has annihilator

$$\prod_{k=1}^c(X-9^k)^{e_{2r,j,k}}.$$

Here $e_{i,j,k}=0$ otherwise, $e_{i,0,i}=1$ for $i\ge0$, $e_{i+1,0,k}=e_{i,0,k}+k$ for $i\ge0$ and $0\le k\le i$, $e_{i,i+1,k+1}=e_{i,0,k}$ for $i\ge0$ and $0\le k\le i$, $e_{i,i,k}=e_{i,i-1,k}+k$ for $i\ge1$ and $1\le k\le i$, and $e_{i,j,k}=e_{i-1,j,k}+k$ for $i>j$, $1\le j\le i$, and $1\le k\le i$. The conjecture gives explicit factorization and recursion data for the rows, but the source says that explicit forms for all entries, or equivalently explicit initial-condition patterns, are not known. It is verified there for rows $0\le r\le7$, and the strong form implies the weak form.