The differential string trivialization conjecture for Witten Pfaffian lines

Let $(M_{8\ell+2},g_M^{\mathrm{spin}})$ be a closed differential spin manifold, and let $g_M^{\mathrm{string}}$ be a lift of its spin structure to a differential string structure. The associated Witten Pfaffian line is denoted by $\operatorname{Pf}(\operatorname{Wit}(M,g_M^{\mathrm{spin}}))$. The differential string trivialization conjecture. Such a lift gives a trivialization of this Pfaffian line up to multiplication by $\exp(i(\operatorname{MF}_{4\ell+2})_{\mathbb{R}})$, so that the fractional reduced eta invariant is well-defined in

$$\frac{\mathbb{R}((q))}{(\operatorname{MF}_{4\ell+2})_{\mathbb{R}}+2\mathbb{Z}((q))}.$$

In terms of this invariant, the differential pushforward is given by the eta-invariant formula stated in the source. The conjecture addresses the failure of canonical Pfaffian-line trivializations for manifolds with boundary and is used to formulate the differential $\operatorname{TMF}$ pushforward.