Burghelea–Haller conjecture on complex Ray–Singer and Farber–Turaev torsions

Let $M$ be a closed oriented manifold of odd dimension $d=2n-1$, let $E$ be a flat complex vector bundle over $M$ with flat connection $\nabla$, and let $b$ be a non-degenerate symmetric bilinear form on $E$. Let $\varepsilon$ be an Euler structure represented by a non-vanishing vector field $X$, let $o$ be a cohomological orientation, and fix a Riemannian metric $g^M$. If $\Psi(g^M)\in\Omega^{d-1}(TM\backslash\{0\})$ is the Mathai–Quillen form, set

$$\gamma_{\varepsilon}=\gamma_{\varepsilon}(g^M):=X^*\Psi(g^M).$$

Burghelea–Haller conjecture. The Burghelea–Haller quadratic form satisfies

$$\tau^{\operatorname{BH}}_{b,\gamma_{\varepsilon},\nabla}\bigl(\rho_{\varepsilon,o}(\nabla)\bigr)=1.$$

This is a precise odd-dimensional formulation of the conjecture that the complex Ray–Singer torsion is, for a suitable choice of form, essentially the square of the Farber–Turaev torsion. Braverman and Kappeler proved a weak version: the displayed quantity is locally constant in $\nabla$ and has absolute value $1$, while the asserted equality to $1$ is the stronger conjectural statement.