In the cohomology or the group cohomology theory, suppose $\mu_1$ and $\mu_2$ are coboundaries of arbitrary dimensions, $$ \mu_1=\delta \eta_1 $$ $$ \mu_2=\delta \eta_2 $$ where $\eta_1$ and $\eta_2$ are their lower dimensional split cochains.
Could we prove that the higher cup 1 product is also a coboundary? $$ \mu_1 \cup_1 \mu_2=(\delta \eta_1)\cup_1 (\delta \eta_2)=\delta(\beta)? $$
If so, how do we write this $\beta$ explicitly?
Is a Higher cup-1 product of coboundaries also a coboundary?