Existence of a surgery curve with vanishing Levine–Tristram signature

Let $K$ be a slice knot and let $F$ be a genus-one Seifert surface for $K$. A Levine–Tristram signature function is the signature function associated to a knot, denoted in the source by $\sigma_K:S^1\to\mathbb{Z}$. Signature-vanishing surgery-curve conjecture. There exists a surgery curve on $F$ with vanishing Levine–Tristram signature function. The paper then gives counterexamples: some slice knots with genus-one Seifert surfaces have no surgery curve with vanishing signature function, so the conjecture is refuted.