Failure of Nehari's Theorem for Multiplicative Hankel Forms in Schatten Classes II

We prove that for any $p>4$ we can find a Schatten class Hankel operator ${\text{Ha}}_{\phi }$ defined on the Hardy space of the infinite torus $H^2({\mathbb{T}}^{\mathrm{\infty }})$ that belongs to the Schatten class $S^p$ but does not admit a bounded symbol, extending the previously known range $p>(1-\log \pi \mathrm{/}\log 4{)}^{-1}=\mathrm{5.7388...}$.