No nonzero product vector lies in the range of $Z = J(\Phi_2 \circ \Phi_1)$

For the Choi matrix $Z = J(\Phi_2 \circ \Phi_1)$ of the composition of the two explicit maps of the dimension-ten pair, the range of $Z$ contains no nonzero product vector: whenever $Z w = u \otimes v$ with $u, v \in \mathbb C^{10}$, one of $u, v$ is zero. The source proves this (Proposition 6.1 and Proposition 7.2) from twenty projective directions $[x]$ in $\mathbb C^4$ at which the pencil map $L$ has singular output: each gives a product vector $\hat x \otimes \hat x$ in the kernel of $Z$, so a product vector in the range would yield two nonzero homogeneous quadratics in four variables whose zero sets cover all twenty directions, while an exact Galois-theoretic certificate shows every nonzero quadratic vanishes on at most nine of them.