Inverse-shift conjecture for almost regular rectangular Birkhoff schemes

Let $(A,S)$ be a Birkhoff interpolation scheme, with $S$ a lower set, and fix non-negative integers $p$ and $q$. An inverse shift moves elements of $A$ downwards and to the left within $S$, as defined in the paper. The notation $R^{p,q}$ denotes the rectangular-node construction applied to a lower set $R$. Inverse-shift conjecture. If $(A,S)$ is almost regular with respect to $(p,q)$-rectangular sets of nodes, then an inverse shift of $A$ in $S$ moves $A$ to a lower set $R$, and

$$S=R^{p,q}.$$

This conjecture gives a geometric form of the structural claim suggested by the paper's examples; no resolution is supplied in the source.