Cappell–Miller conjecture on torsion and Reidemeister torsion

Let $(E,\nabla)$ be a flat vector bundle over a closed oriented odd-dimensional manifold $M$. Let $T_\nabla$ denote its Cappell–Miller torsion, let $E^*$ be the dual bundle with dual connection $\nabla^*$, let $D:\operatorname{Det}(H^\bullet(M,E))\to\operatorname{Det}(H^\bullet(M,E^*))$ be the Farber–Turaev duality operator, and let $\rho^{\operatorname{R}}(\nabla\oplus\nabla^*)$ be the Reidemeister torsion of $E\oplus E^*$. Let $z\in\mathbb{N}$ be the integer defined in formula (6.5) of Farber–Turaev.

Cappell–Miller conjecture. The Cappell–Miller torsion satisfies

$$(1\otimes D)T_\nabla=(-1)^z\rho^{\operatorname{R}}(\nabla\oplus\nabla^*).$$

The statement relates the analytic Cappell–Miller torsion to the Reidemeister torsion of the self-dual bundle $E\oplus E^*$. The source presents it as equivalent to a reformulation of the Burghelea–Haller conjecture; its resolution is not specified in the supplied text.