  In \cite{Vakil13} Vakil and Wood made several conjectures on the topology of
symmetric powers of geometrically irreducible varieties based on their
computations on motivic zeta functions. Two of those conjectures are about
subspaces of $\Sym^n(\P^1)$. In this note, we disprove one of them thereby
obtaining a counterexample to the principle of Occcam's razor for Hodge
structures; and we prove that the other conjecture, with a minor correction,
holds true.
