On hypercyclic spaces and (common) $\mathscr{U}$-frequently hypercyclic spaces

Let $B$ be an unilateral weighted backward shift on $\ell_p$, $1 \leq p < \infty$, that admits a $\mathscr{U}$-frequently hypercyclic subspace. We prove that $B$ admits such a subspace free of frequently hypercyclic vectors. The proof technique we develop also allows us to prove that $B$ admits a hypercyclic subspace free of $\mathscr{U}$-frequently hypercyclic vectors, and to solve a question posed by B\`es and Menet in 2015 on the existence of common $\mathscr{U}$-frequently hypercyclic subspaces.