Yamaguchi–Yau genus-two mirror conjecture for the quintic threefold

Let $q$ be the degree variable, let $H$ be a formal variable satisfying $H^4=0$, and define the $I$-function, its components $I_0,I_1,I_2,I_3$, the mirror map $Q=q e^{\tau_Q(q)}$ with $\tau_Q(q)=I_1(q)/I_0(q)$, and the generators

$$\mathcal X_k=\frac{d^k}{du^k}\left(\log\frac{I_0}{L}\right),\qquad \mathcal Y_k=\frac{d^k}{du^k}\left(\log\frac{I_0I_{1,1}}{L^2}\right),\qquad \mathcal Z_k=\frac{d^k}{du^k}\left(\log(q^{1/5}L)\right),$$

where $I_{1,1}=1+q\frac{d}{dq}\tau_Q$, $L=(1-5^5q)^{-1/5}$, and $du=L\frac{dq}{q}$. Let $F^{GW}_2(Q)$ denote the genus-two Gromov–Witten generating function of the quintic threefold.

Yamaguchi–Yau genus-two mirror conjecture. The function $F^{GW}_2(Q)$ is given by

$$F^{GW}_2(Q)=\frac{I_0^2}{L^2}\left(\frac{70\mathcal X_3}{9}+\frac{575\mathcal X\mathcal X_2}{18}+\frac{5\mathcal Y\mathcal X_2}{6}+\frac{557\mathcal X^3}{72}-\frac{629\mathcal Y\mathcal X^2}{72}-\frac{23\mathcal Y^2\mathcal X}{24}-\frac{\mathcal Y^3}{24}+\frac{625\mathcal Z\mathcal X_2}{36}-\frac{175\mathcal Z\mathcal Y\mathcal X}{9}+\frac{1441\mathcal Z_2\mathcal X}{48}-\frac{25\mathcal Z(\mathcal X^2+\mathcal Y^2)}{24}-\frac{3125\mathcal Z^2(\mathcal X+\mathcal Y)}{288}+\frac{41\mathcal Z_2\mathcal Y}{48}-\frac{625\mathcal Z^3}{144}+\frac{2233\mathcal Z\mathcal Z_2}{128}+\frac{547\mathcal Z_3}{72}\right).$$

In particular, its leading terms are

$$F^{GW}_2(Q)=-\frac{5}{144}+\frac{575}{48}Q+\frac{5125}{2}Q^2+\frac{7930375}{6}Q^3+O(Q^4).$$

The paper states a main theorem proving this genus-two mirror conjecture, so the claim is solved.