On a result concerning algebraic curves passing through $n$-independent nodes

Let a set of nodes $\mathcal X$ in the plane be $n$-independent, i.e. each node has a fundamental polynomial of degree $n.$ Assume that $\#\mathcal X=d(n,n-3)+3= (n+1)+n+\cdots+5+3.$ In this paper we prove that there are at most three linearly independent curves of degree less than or equal to $n-1$ that pass through all the nodes of $\mathcal X.$ We provide a characterization of the case when there are exactly three such curves. Namely, we prove that then the set $\mathcal X$ has a very special construction: either all its nodes belong to a curve of degree $n-2,$ or all its nodes but three belong to a (maximal) curve of degree $n-3.$
This result complements a result established recently by H. Kloyan, D. Voskanyan, and H. Hakopian. Note that the proofs of the two results are completely different.