On the cohomology of certain subspaces of ${Sym}^n({\mathrm{\P }}^1)$ and Occam's razor for Hodge structures

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 ${\text{Sym}}^n({\mathrm{\P }}^1)$. 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.