Integrality conjecture for open-closed mirror maps

Let $x_0,x_1,\dots,x_N$ be variables for a system of Picard–Fuchs equations arising from charge vectors describing brane geometry in a Calabi–Yau geometry in the large-volume phase. Suppose the system has a holomorphic solution $g_0(x_0,\dots,x_N)$ with $g_0(0,\dots,0)=1$ and logarithmic solutions

$$g_1^{(i)}(x_0,\dots,x_N)=\log x_i\,g_0(x_0,\dots,x_N)+h_1^{(i)}(x_0,\dots,x_N),\qquad i=0,1,\dots,N,$$

where each $h_1^{(i)}$ is holomorphic and satisfies $h_1^{(i)}(0,\dots,0)=0$. Define the open-closed mirror-map coordinates by

$$q_i=\exp\left(g_1^{(i)}/g_0\right)=x_i\exp\left(h_1^{(i)}/g_0\right),\qquad i=0,1,\dots,N.$$

Integrality conjecture. The Taylor series of $q_0,q_1,\dots,q_N$ in $x_0,x_1,\dots,x_N$ have integer coefficients, and the inverse Taylor series likewise have integer coefficients.

This extends the integrality phenomenon known for closed mirror maps and observed for open-closed mirror maps in noncompact Calabi–Yau threefolds to compact Calabi–Yau manifolds and further noncompact examples.