  We study the reciprocal variety to the linear space of symmetric matrices
(LSSM) of catalecticant matrices associated with ternary quartics. With
numerical tools, we obtain 85 to be its degree and 36 to be the ML-degree of
the LSSM. We provide a geometric explanation to why equality between these two
invariants is not reached, as opposed to the case of binary forms, by
describing the intersection of the reciprocal variety and the orthogonal of the
LSSM in the rank loci. Moreover, we prove that only the rank-$1$ locus, namely
the Veronese surface $\nu_4(\mathbb{P}^2)$, contributes to the degree of the
reciprocal variety.
