Kitaev's determinant conjecture for Fredholm commutators

Let $\mathcal H$ be a Hilbert space and let $\mathcal S_1$ denote the Schatten trace class. For invertible operators $A,B\in\mathcal L(\mathcal H)$, assume

$$(A-I)(B-I),(B-I)(A-I)\in\mathcal S_1.$$

Kitaev's determinant conjecture. Under these assumptions, the Fredholm determinant satisfies

$$\det\left(ABA^{-1}B^{-1}\right)=1.$$

This conjecture asks when the finite-dimensional determinant identity for commutators extends to infinite-dimensional operators. The surrounding discussion attributes the claim to Kitaev's formal computation and explains that it would imply quantization of the trace of suitable trace-class commutators; its resolution is not established in the supplied text.