The Atiyah-Sutcliffe conjecture and $E_n$-algebras

We show that a certain conjecture by Atiyah and Sutcliffe implies the existence of an $E_3$-algebra (respectively $E_2$-algebra) structure on the disjoint union of all complex (respectively real) full flag manifolds modulo symmetric groups. Moreover, we show that these structures are liftings of exotic $E_3$ (respectively $E_2$) structures on the free $E_{\mathrm{\infty }}$-algebras on $BU(1)+$ (respectively $BO(1)+$), that do not extend to $E_4$ (respectively $E_3$) structures. We also provide some (co)homological calculations supporting the conjecture.