Fox's trapezoidal conjecture for alternating knots

Let $K$ be an alternating knot, and write its Alexander polynomial as $\Delta_K(t)=\sum_i a_i t^i$. A sequence is trapezoidal if the absolute values of its terms increase, possibly remain constant on a plateau, and then decrease. Fox's trapezoidal conjecture. The absolute values $|a_i|$ form a trapezoidal sequence. A stronger variant asserts that the non-plateau portions are log-concave, satisfying

$$a_i^2 > |a_{i-1}|\cdot |a_{i+1}|.$$

Murasugi established fundamental algebraic properties for alternating knots, but structural proofs of the trapezoidal condition remain open for general alternating families; the stronger log-concavity variant is likewise not established here.