Bley–Burns–Hahn conjecture on Galois-Gauss sums

Let $L/K$ be a weakly ramified Galois extension of number fields of odd degree. Write $\mathfrak{a}_{L/K}$ for the canonical pre-image associated with the stable-isomorphism class of the square root of the inverse different, and let $\mathfrak{c}_{L/K}$ denote the idelic twisted unramified characteristic attached to $L/K$, both viewed in $K_0(\mathbb{Z}[G],\mathbb{Q}[G])$, where $G=\operatorname{Gal}(L/K)$. Bley–Burns–Hahn conjecture. One has

$$\mathfrak{a}_{L/K}=\mathfrak{c}_{L/K}.$$

This conjecture refines the study of the Galois module structure of the square root of the inverse different by comparing a canonical relative $K$-theory class with an idelic twisted unramified characteristic. It is supported by numerical computations and is the main conjecture investigated in the paper; the supplied text gives no resolution.