Cup Product in Bounded Cohomology of the Free Group

The theory of bounded cohomology of groups has many applications. A key open problem is to compute the full bounded cohomology $H_b^n(F,R)$ of a non-abelian free group $F$ with trivial real coefficients. It is known that $H_b^n(F,R)$ is trivial for $n=1$ and uncountable dimensional for $n=2,3$, but remains unknown for any $n\ge 4$. For $n=4$, one may construct classes by taking the cup product $\alpha \cup \beta \in H_b^4(F,R)$ between two $2$-classes $\alpha ,\beta \in H_b^2(F,R)$. However, we show that all such cup products are trivial if $\alpha$ and $\beta$ are classes induced by the quasimorphisms defined by Brooks or Rolli.