  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 $n \geq 4$. 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 $\alpha, \beta \in H^2_b(F, R)$. 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.
