The contact-star formula for up to four general fat points

Let $s\le 4$ and let $m_i$ be the multiplicities of $s$ general fat points. For each $i$, take a contact star configuration formed by $m_i+1$ lines tangent to the same conic. The contact-star $h$-vector conjecture. The $h$-vector of the scheme of the $s$ general fat points is equal to the $h$-vector of the union of these $s$ contact star configurations. The claim is suggested by the preceding theorem and computational experiments; its status is not resolved in the supplied source.