Weak equivalences in the localized model structure on all simplicial sets

Determine whether, in the left Bousfield localized model structure L_{M^{-1}Z}SSets on all simplicial sets obtained by localizing at the set B_{M^{-1}Z} = { Sing(S^n) → Sing(S^n_{M^{-1}Z}) | n ≥ 2 }, the weak equivalences coincide with the M-local homotopy equivalences (i.e., those maps whose geometric realizations are M-local homotopy equivalences of spaces as in Definition 9.1).

Background

To extend rational (or M-local) homotopy theory beyond 1-connected spaces, the authors construct model structures on all simplicial sets by left Bousfield localization at maps induced from S^n → S^n_{M^{-1}Z}. They can characterize the fibrant objects (the M-local simplicial sets) using function complexes.

However, due to the lack of a straightforward universal property for localization outside the 1-connected setting, their method does not establish that the weak equivalences in the localized model structure L_{M^{-1}Z}SSets coincide with M-local homotopy equivalences. Clarifying this equivalence would complete the identification of weak equivalences analogous to the 1-reduced case.