Cobordism genera can often be refined to $E_\infty$-orientations in the sense of Ando-Blumberg-Gepner-Hopkins-Rezk:

1) the mod 2 Euler characteristic $MO\to H\mathbb{F}_2$;

2) the $\widehat A$-genus $MSpin\to KO$ (Ando-Hopkins-Rezk, Joachim);

3) the Todd genus $MSpin^c\to K$ (Joachim);

4) the Witten genus $MString\to tmf$ (Ando-Hopkins-Rezk).

Now, the signature (and Arf invariant, etc.) lifts to an $\mathbb{L}$-theory orientation for $PL$-bundles (e.g. Ranicki), and $MPL$ and $\mathbb{L}$ are both $E_\infty$-ring spectra.

My question is the following: is it known whether this orientation has an $E_\infty$ refinement? If the answer is yes, I would appreciate a reference.