Product-expansion conjecture for two-variable open-closed mirror maps

Let $q_0,q_1$ and $Q_0,Q_1$ be the two systems of mirror-map coordinates related by the integral power-series changes of variables arising in the two-variable open-closed mirror-map setting. For $i=0,1$, write the corresponding hypergeometric solution in the form

$$g_0(z_0,z_1)=1+\sum_{m_0+m_1>0}c_{m_0,m_1}q_0^{m_0}q_1^{m_1}=1+\sum_{m_0+m_1>0}C_{m_0,m_1}Q_0^{m_0}Q_1^{m_1}.$$

Product-expansion conjecture. There are integers $\alpha^{(i)}_{m_0,m_1}$ and $\beta^{(i)}_{m_0,m_1}$ such that

$$Q_i=q_i\prod_{m_0+m_1\geq1}\left(1-q_0^{m_0}q_1^{m_1}\right)^{\alpha^{(i)}_{m_0,m_1}},$$ $$q_i=Q_i\prod_{m_0+m_1\geq1}\left(1-Q_0^{m_0}Q_1^{m_1}\right)^{\beta^{(i)}_{m_0,m_1}},$$

for $i=0,1$.

This is presented as a consequence of the preceding integrality conjecture: integrality of the mirror maps and their inverses motivates the stronger Euler-product-type expansions, but the source does not report a resolution.