Trapezoidal conjecture for Conway polynomial coefficients of two-bridge links

Let $m$ be the degree parameter of the Conway polynomial, let $c_m$ be its leading coefficient, and let $f_j$ denote the Fibonacci-basis polynomials. Write the Conway polynomial of a two-bridge link or knot as

$$\nabla_m=c_m\Bigl(\sum_{i=0}^{\lfloor m/2\rfloor}(-1)^i\alpha_i f_{m+1-2i}\Bigr),\qquad \alpha_0=1.$$

Trapezoidal conjecture. There exists an integer $n\leq\lfloor m/2\rfloor$ such that

$$0\leq\alpha_0\leq\alpha_1\leq\cdots\leq\alpha_n,\qquad \alpha_n\geq\alpha_{n+1}\geq\cdots\geq\alpha_{\lfloor m/2\rfloor}\geq0.$$

The conjecture was motivated by computations of the Conway polynomials of all $131\,839$ two-bridge links and knots with at most 20 crossings. If true, it would imply the stated convexity property for the coefficients of the Alexander polynomial of every two-bridge knot.