Conjectural structure of lines in GC_n sets

Assume the Gasca–Maeztu conjecture holds for all degrees up to $\nu$. Let $\mathcal X$ be a $GC_n$ set with $n\leq\nu$, and let $\ell$ be a line passing through exactly $k$ nodes of $\mathcal X$. Write $\mathcal X_\ell$ for the nodes of $\mathcal X$ that use $\ell$ in their fundamental polynomials. A maximal line is a line containing $n+1$ nodes of $\mathcal X$.

Conjecture concerning $GC_n$ sets. There is an integer $s$ with $2k-n-1\leq s\leq k$ such that

$$|\mathcal X_\ell|=\binom{s}{2}.$$

Moreover, for every maximal line $M$ of $\mathcal X$:

1. $|M\cap\mathcal X_\ell|=0$ if $M\cap\ell\notin\mathcal X$, or if there is another maximal line $M'$ such that $M\cap M'\cap\ell\in\mathcal X$;
2. $|M\cap\mathcal X_\ell|=s-1$ for every remaining maximal line.

This is a structural conjecture about how nodes using a given line are distributed among maximal lines. Its validity is conditional on the Gasca–Maeztu conjecture through degree $\nu$; the source provides no resolution.