Nica's proper affine action conjecture for mapping class groups

Let $G$ be the mapping class group of a surface. A proper affine isometric action on an $L_p$-space is an affine isometric action of $G$ on some $L_p$-space whose orbit maps are proper.

Nica's conjecture. The mapping class group of a surface admits a proper affine isometric action on an $L_p$-space for some sufficiently large

$$p=p(G)\in[2,\infty).$$

This conjecture concerns the existence of proper affine isometric actions on uniformly convex Banach spaces and is stated in the source as open.