As a follow-up to the earlier post https://mathoverflow.net/q/448599/28128, I have not been able to prove the analogous result for a wedge product $a\wedge b\wedge c\wedge d$ of 2-forms in $\mathbb R^8$.  If one of the forms occurs twice then one can use an endomorphism $M_{\alpha^{\wedge2}}$ of $\bigwedge^2(\mathbb R^8)$ as in the earlier post, so as to obtain the tight bound $4!$, but otherwise I can only find an upper bound of $64$ (which is better than $8!/2^4=2520$).  What is the maximal value one obtains for a product of four 2-forms of unit comass in $\mathbb R^8$?