Hirasawa–Murasugi conjecture on dihedral twisted Alexander polynomials

Let $K$ be a knot and let

$$\phi:\pi_1(S^3-K)\to D_\ell\to GL(\ell,\mathbb C)$$

be a non-abelian linear representation, where $D_\ell$ is the dihedral group of order $2\ell$ and $\ell$ is an odd prime. Hirasawa–Murasugi conjecture. There is an integer Laurent polynomial $f(t)$ such that

$$\widetilde\Delta_{K,\phi}(t)=\frac{\Delta_K(t)}{t-1}f(t)f(-t)$$

and

$$f(t)\equiv\left(\frac{\Delta_K(t)}{t+1}\right)^{(\ell-1)/2}\pmod{\ell}.$$

The paper studies this conjecture for 2-bridge knots and confirms it for several families, including torus knots and genus-one knots; the general assertion stated here is not resolved by the supplied context.