A remark on Waring decompositions of some special plane quartics

Motivated by questions on tensor rank, this work concerns the following unexpected result concerning Waring decompositions of plane quartics containing a double line, along with some preparatory and additional remarks. 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 the projective plane intersect x=0 at seven distinct points, then the line x=0 is (possibly improperly) tangent to the conic q=0.