$\DeclareMathOperator\Conf{Conf}$Let $M$ be a manifold, and $\Conf_n M$ the ordered configuration space of n points on $M$. The symmetric group $S_n$ acts by permuting the points.

Is there a simple description of the homotopy fixed points of this action ? In particular I wonder if this is not empty. For $M = \mathbb R^2$ it seems to me that the existence of a coherent $S_n$-equivariant point on $\Conf_n M$ would imply that there is a section $S_n \to B_n$, the braid group on $n$ strands, because there would be a choice of a (homotopy class of) path from $(x_1,...,x_n)$ to $(x_\sigma(1),...,x_\sigma(n))$ that commutes with composition of permutations. And I know that it doesn't exist.