The convex-concave conjecture for Alexander polynomials of two-bridge knots

Let

$$\Delta(t)=a_0-a_1(t+t^{-1})+a_2(t^2+t^{-2})-\cdots+(-1)^n a_n(t^n+t^{-n})$$

be the Alexander polynomial of a two-bridge knot. Convex-concave conjecture. There exists an integer $k\leq n$ such that $(a_0,\ldots,a_k)$ is convex and $(a_k,\ldots,a_n)$ is concave.

The source states this as a consequence that would follow from the trapezoidal conjecture. It notes that the sequence of Alexander coefficients is not, in general, convex, so the asserted change from convexity to concavity is the relevant remaining pattern.