The zero-volume conjecture for knot exteriors and Mahler measure

Let $K$ be a knot, let $\mathcal{M}_{K}$ denote its exterior, let $A_K$ be its $A$-polynomial, let $\mathrm{Vol}(\mathcal{M}_K)$ be the sum of the volumes of the hyperbolic pieces in its JSJ-decomposition, and let $\mathrm{m}(A_K)$ denote the Mahler measure of $A_K$. Zero-volume conjecture. A knot exterior $\mathcal{M}_{K}$ has

$$\mathrm{Vol}(\mathcal{M}_{K})=0$$

if and only if

$$\mathrm{m}(A_K)=0.$$

Equivalently, $\mathcal{G}_{0}=\mathfrak{M}_{0}$. Graph knots are exactly the knots whose exteriors have zero hyperbolic volume, and the preceding theorems establish the forward direction; the converse remains open.