Minimal Hilbert function for the first infinitesimal neighborhood of a k-configuration

Let $(n_1,\ldots,n_r)$ be a type defining a generic Hilbert function, and construct the associated $k$-configuration $C_{\underline h}$ from $r+1$ lines by taking the pairwise intersection points and adding the required arbitrary points on the lines. Let $\underline h^{\min}$ denote the minimal Hilbert function for first infinitesimal neighborhoods of point sets with the given Hilbert function.

Linear-configuration conjecture. The Hilbert function of $C_{\underline h}$ is $\underline h^{\min}$.

This is presented as a generalization of the preceding conjecture from configurations of type $(1,2,\ldots,t-1,r)$. The paper provides the construction and motivating examples, but no proof is given in the supplied text.