Alexander polynomial conjecture for the monodromy of associated surface singularities

Let $C_0$ and $C_1$ be the curves and let $g_0$ and $g_1$ be the polynomial function germs associated with them. Write $V(g_j)$ for the zero locus of $g_j$ and let the monodromy of $g_j$ act on its Milnor fiber. The Alexander polynomial conjecture. If the Alexander polynomials of $C_0$ and $C_1$ are different, then the Jordan form of the monodromy of $g_0$ is different from that of $g_1$, and therefore the surface germs $(V(g_0),\mathbf{0})$ and $(V(g_1),\mathbf{0})$ have distinct embedded topologies in $\mathbb{C}^3$. This would strengthen the paper's theorem from a $\mu$-Zariski pair to an ordinary Zariski pair, and is stated without necessarily assuming that the singularities of $C_0$ and $C_1$ are Newton non-degenerate.