The sign-pattern conjecture for coefficients of mirror maps

Let $N_1,N_2,\ldots,N_k$ be positive integers, all at least $2$, and let ${\bf z}_{\bf N}(q)$ be the corresponding mirror map. Sign-pattern conjecture.

1. If $\Phi_{\mathbf N}\le3$, the Taylor coefficients of ${\bf z}_{\bf N}(q)$ have alternating signs.
2. If $\Phi_{\mathbf N}=4$, the coefficients of $q$ and $q^3$ in the Taylor series of ${\bf z}_{\bf N}(q)$ are positive, while all other coefficients are negative, except the constant coefficient, which vanishes.
3. If $\Phi_{\mathbf N}\ge5$, the coefficient of $q$ in the Taylor series of ${\bf z}_{\bf N}(q)$ is positive, while all other coefficients are negative, except the constant coefficient, which vanishes.

These sign assertions concern the Taylor expansion at $q=0$. The paper notes that part (i) is known in several specific cases and asymptotically for general parameters, while the full sign pattern remains conjectural.