Identification of the index-theoretic and homotopy-theoretic MU-local systems

Prove that the Thom spectrum associated to the homotopy-theoretically defined MU-local system \(\Sigma^{-\dim Q}\widetilde V\otimes\epsilon\) on \(LQ\) is equivalent to the Thom spectrum associated to the index-theoretic MU-local system \(\Sigma^{-\dim Q}V\otimes\epsilon\), namely establish \((LQ)^{\Sigma^{-\dim Q}\widetilde V\otimes\epsilon}\simeq(LQ)^{\Sigma^{-\dim Q}V\otimes\epsilon}\).

Background

The paper defines a candidate local system $\widetilde V$ by applying the free-loop construction to the stable tangent bundle classifying map $[TQ]:Q\to\mathrm{BO}$, then mapping through $$L\mathrm{BO}\to\Omega\mathrm{BO}\simeq\mathrm O\to\mathrm{BGL}_1(\mathrm{MU})$$. This construction is intended to provide the homotopy-theoretic counterpart of the index-theoretically constructed local system $V$.

The conjecture asks whether the two resulting Thom spectra agree after including the dimension shift and the auxiliary local system $\epsilon$. Establishing this would give the desired homotopy-theoretic interpretation of the MU-local system used in the MU-Viterbo isomorphism.