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

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