Mock modularity conjecture for log Gromov–Witten series on the mirror to $\mathbb{P}^2$

Let $(Y,D)$ be the rational elliptic surface mirror to $\mathbb{P}^2$. For each $r\in\mathbb{Z}_{>0}$ and each integer $c_1$ with $-r<c_1\leq 0$, let $h^{GW}_{r,c_1}(\tau)$ be the corresponding logarithmic Gromov–Witten generating series. Mock modularity conjecture. The vector

$$\bigl(h^{GW}_{r,c_1}(\tau)\bigr)_{-r<c_1\leq 0}$$

is a vector-valued mock modular form of weight $-\frac{3}{2}$ and depth $r$. The statement extends the proved cases $1\leq r\leq 3$ and is motivated by the mock modularity of the corresponding Vafa–Witten generating series.