A remark on Waring decompositions of some special plane quartics

This work concerns Waring decompositions of a certain kind of plane quartics of high rank. The main result is the following. Let x, l_1, …, l_7 be linear forms and q a quadratic form on a vector space of dimension 3. If x^2q=l_1^4+…+l_7^4 and the lines l_1=0, …, l_7=0 in P^2 intersect x=0 in seven distinct points, then the line x=0 is (possibly improperly) tangent to the conic q=0.