Bley–Burns–Hahn's relative K-theory conjecture for weakly ramified extensions

Let $L/K$ be a weakly ramified Galois extension of number fields of odd degree, with $G=\operatorname{Gal}(L/K)$. Let $\mathfrak{a}_{L/K}$ and $\mathfrak{c}_{L/K}$ be the canonical elements in the relative group $K_0(\mathbb{Z}[G],\mathbb{Q}[G])$ introduced by Bley, Burns and Hahn, where $\mathfrak{a}_{L/K}$ measures the difference between the invariant arising from $\mathcal{A}_{L/K}$ and the second Adams-operator-twisted Galois–Gauss sums. Bley–Burns–Hahn's conjecture. If $L/K$ is any weakly ramified Galois extension of number fields of odd degree, then, in $K_0(\mathbb{Z}[G],\mathbb{Q}[G])$, one has

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

The conjecture is motivated by torsion and functoriality results for $\mathfrak{a}_{L/K}$ and by numerical computations. The supplied source does not report a resolution.