Is there a chance a "continuity" argument could work? For instance, consider the product of vectors $abcde$, and lets say only $c$ is null. Choose null $c'$ such that $c\cdot c' = 1$; then $c + \epsilon c'$ is non-null for all non-zero $\epsilon \in \mathbb R$. So then $ab(c+\epsilon c')de$ is a product of at most three vectors for all $\epsilon\not= 0$. What happens as $\epsilon \to 0$?

I tried to do an example by hand... but clearly I'm incapable of doing this without constantly making mistakes. I was at least able to stumble upon