Polynomial-growth conjecture at minimal Alexander roots

Let $K$ be a knot, and let $\Delta(K;t)$ be its Alexander polynomial. Define

$$\Lambda(K):=\{z\in\mathbb{C}\mid\Delta(K;\exp z)=0\}.$$

Choose $c\in\Lambda(K)$ such that $|c|=\min\{|z|\mid z\in\Lambda(K)\}$. Polynomial-growth conjecture. The sequence

$$\left\{J_N\bigl(K;\exp(c/N)\bigr)\right\}_{N=2,3,\dots}$$

grows polynomially, and

$$\lim_{N\to\infty}J_N\left(K;\exp\left(\frac{tc}{N}\right)\right)=\frac{1}{\Delta\bigl(K;\exp(tc)\bigr)}$$

for $0\le t<1$. This conjecture describes the transition from the Melvin–Murakami asymptotic formula away from zeros of the Alexander polynomial to polynomial growth at a zero of minimal modulus; its general validity remains open.