MNOP rationality and DT/GW correspondence conjecture

Let $X$ be a smooth projective Calabi–Yau threefold. For a curve class $\beta\beta$ and the rank-one Donaldson–Thomas series $I_\beta(X)$, let $I_0(X)$ denote the degree-zero series, and let $\mathrm{GW}_{g,\beta}(X)$ denote the Gromov–Witten invariant in genus $g$ and class $\beta$. MNOP conjecture. (i) The quotient series $I_{\beta}(X)/I_0(X)$ is the Laurent expansion of a rational function of $q$, invariant under $q\beta 1/q$. (ii) After the variable change $q=-e^{iclambda}$, one has

$$\sum_{\beta \ge 0} \frac{I_{\beta}(X)}{I_0(X)}t^{\beta} = \exp\left( \sum_{g\ge 0, \beta>0} \mathrm{GW}_{g, \beta}(X)\lambda^{2g-2} t^{\beta} \right).$$

The rationality part is stated in the paper as proved, while the DT/GW generating-series identity is the associated correspondence motivating the conjecture; its precise resolution is not established by the supplied excerpt.