The seven-Pfaffian conjecture for the rank-one catalecticant image

Let $C_1$ denote the rank-one locus of the relevant catalecticant variety, let $A$ be a point on $C_1$, and let $\phi(A)$ be its image under the map $\phi$. Seven-Pfaffian conjecture. If $A$ is a point on $C_1$, then $\phi(A)$ is defined by the $7$ cubic Pfaffians of a $7 \times 7$ skew-symmetric matrix. This would extend the known description for points on $C_2$, whose images are defined by the cubic Pfaffian of a $6 \times 6$ skew-symmetric matrix; the claim is presented as a dimension-count prediction and its resolution is not supplied here.