Signature-controlled unimodality for alternating-knot Alexander polynomials

Let

$$\Delta_K(t)=\sum_{j=0}^{2n}(-1)^jc_jt^{2n-j},\qquad c_j>0,$$

be the Alexander polynomial of an alternating knot, and let $\sigma(K)$ be its signature. Signature-controlled coefficient conjecture. If $\sigma(K)=0$, then

$$c_0<c_1<\dots<c_n>c_{n+1}>c_{n+2}>\dots>c_{2n}.$$

More generally, if $\sigma(K)=2k$, then there is an integer $m\leq k$ such that

$$c_0<c_1<\dots<c_{n-m-1}<c_{n-m}=\dots=c_{n+m}>c_{n+m+1}>\dots>c_{2n}.$$

The source says this is plausible for 2-bridge knots but false for non-alternating knots.