Weakly cyclic vectors with a given modulus

Let $H^p$ be the Hardy space in the polydisc. Denote by $\mathcal P$ the set of all holomorphic polynomials. A vector $f\in H^p$ is called weakly cyclic if the product $f\mathcal P$ is weakly dense in $H^p$, $0<p<1$. We construct weakly cyclic vectors with a prescribed lower semicontinuous modulus of the boundary values.