The generalized evaluation conjecture for twisted Alexander polynomial quotients

Suppose there is an epimorphism $\varphi:G(K(r))\to G(K(r_0))$, and let $\rho$ be a canonical parabolic representation of $G(K(r_0))$ into $SL(2,\mathbb C)$. Let $\widetilde{\Delta}_{\rho\varphi,K(r)}(t)$ and $\widetilde{\Delta}_{\rho,K(r_0)}(t)$ be the associated twisted Alexander polynomials, and define $\lambda_{\rho,K(r)}(t)$ by

$$\widetilde{\Delta}_{\rho\varphi,K(r)}(t)=\lambda_{\rho,K(r)}(t)\widetilde{\Delta}_{\rho,K(r_0)}(t).$$

Generalized evaluation conjecture.

$$\lambda_{\rho,K(r)}(1)=1,$$

and

$$\lambda_{\rho,K(r)}(-1)=\mu^2$$

for some $\mu\in\mathbb Z[s_{r_0}]$.

This is proposed as a generalization of the paper's main theorem. The source gives an example satisfying both evaluations, but does not establish the conjecture in general.