The regular-representation congruence for twisted Alexander polynomials of knots

Let $p$ be the prime appearing in the dihedral group $D_p$, let $K$ be a knot with group $G(K)$, and let

$$\rho: D_p \ltimes C_p \to GL(2p^2,\mathbb{Z})$$

be the regular representation of $D_p \ltimes C_p$. Suppose that there is a surjective homomorphism

$$f:G(K)\to D_p\ltimes C_p.$$

Regular-representation congruence. The twisted Alexander polynomial satisfies

$$\Delta_K^{\rho\circ f}(t)\equiv\left(\frac{\Delta_K(t)}{t-1}\cdot\frac{\Delta_K(-t)}{t+1}\right)^{p^2}\pmod p.$$

This conjecture generalizes the preceding proposition concerning the group $D_p\ltimes C_p$ and predicts a uniform modulo-$p$ formula for the twisted Alexander polynomial associated with its regular representation. The supplied text gives no evidence resolving the conjecture, so its status remains open.