The real Atiyah–Sutcliffe conjecture for Fulton–MacPherson compositions

Let $FM_2(n)$ be the Fulton–MacPherson operad space, let $F_{k_i}^{\mathbb{R}}\subseteq D_{k_i}(BO(1))$ consist of projectively independent real labels, and let $\nu_{n:k_1,\dots,k_n}^{\mathbb{R}}$ be the induced composition map. Put $K=\sum_{i=1}^n k_i$. The real Atiyah–Sutcliffe $E_2$ conjecture.

$$\nu_{n:k_1,\dots,k_n}^{\mathbb{R}}\left(FM_2(n)\times\prod_{i=1}^nF_{k_i}^{\mathbb{R}}\right)\subseteq F_K^{\mathbb{R}}.$$

Equivalently, the $K$ elements of $BO(1)=\mathbb{RP}^{\infty}$ produced by this map are always projectively independent. The statement is the real analogue of the preceding conjecture and is the condition needed for the corresponding $E_2$-structure; its general resolution is not given.