  It is an open problem whether Kirk's $\sigma$ invariant is the complete
obstruction to a link map $S^2\cup S^2\to S^4$ being link homotopically
trivial. With the objective of constructing counterexamples, Li proposed a link
homotopy invariant $\omega$ that is defined on the kernel of $\sigma$ and also
obstructs link nullhomotopy. We show that $\omega$ is determined by $\sigma$,
and is a strictly weaker invariant.
