Bershadsky–Cecotti–Ooguri–Vafa quintic mirror genus-one and BCOV conjectures

Let $\pi\colon\mathcal W\to\mathbb P^{1}$ be a family of quintic mirror threefolds. Let $N_g(d)$ be the genus-$g$ Gromov–Witten invariant of degree $d$ of a general quintic threefold in $\mathbb P^{4}$. Under the mirror map, let $F_{1,A}^{\rm top}(q)$ and $F_{1,B}^{\rm top}(\psi)$ be the functions defined in the source, and let $\tau_{\rm BCOV}(W_\psi)$ be the BCOV invariant. Let $\varXi_\psi$ be the specified holomorphic $3$-form, and let $\|\cdot\|$ be the specified Hermitian metric on $(\pi_*K_{\mathcal W/\mathbb P^{1}})^{\otimes62}\otimes(T\mathbb P^{1})^{\otimes3}|_{\mathbb P^{1}\setminus\mathcal D}$. Bershadsky–Cecotti–Ooguri–Vafa quintic mirror conjectures. Both the following assertions should hold:

$$q\frac{d}{dq}\log F_{1,A}^{\rm top}(q) = \frac{50}{12} - \sum_{n,d=1}^{\infty}N_{1}(d)\, \frac{2nd\,q^{nd}}{1-q^{nd}} - \sum_{d=1}^{\infty}N_{0}(d)\, \frac{2d\,q^{d}}{12(1-q^{d})}.$$

and, near $\psi=\infty$,

$$\tau_{\rm BCOV}(W_{\psi}) = {\rm Const.}\, \left\|\psi^{-62}(\psi^{5}-1)^{\frac12}(\varXi_\psi)^{62}\otimes\left(\frac{d}{d\psi}\right)^{3}\right\|^{\frac23} = {\rm Const.}\, \left\|\frac{1}{F_{1,B}^{\rm top}(\psi)^{3}}\left(\frac{\varXi_\psi}{y_0(\psi)}\right)^{62}\otimes\left(q\frac{d}{dq}\right)^{3}\right\|^{\frac23}.$$

The source later proves part (B) using its Theorem 11.3, while it does not supply a resolution for part (A); the two parts are therefore separated in status below.