The sign conjecture for the quadratic Gauss sum $G(Q)$

Let $K/F$ be the ramified quadratic extension and let $\chi$ and $\psi$ be the character and additive character above, with $a(\chi)=2t+1$ and $n(\psi)=2l+1$. Define

$$G(Q):=q_K^{-1/2}\sum_{x\in P_K^t/P_K^{t+1}}Q(x),\qquad Q(x):=\chi^{-1}(1+x)(c'^{-1}\psi)(x).$$

Sign conjecture. With the above notations, $G(Q)$ is $\pm1$.

This assertion concerns the sign of the normalized quadratic Gauss sum occurring in the odd-conductor formula for the epsilon factor. The supplied text does not indicate whether the assertion has been proved or remains open.