The coefficient-positivity conjecture for transformed mirror maps

Let $N_1,N_2,\ldots,N_k\ge2$ be integers, let $n$ be a positive integer, and let ${\bf q}_{\bf N}(z)$ be the function associated with the mirror map. For a power series, write $[z^m]$ for the coefficient of $z^m$. Coefficient-positivity conjecture. The coefficient

$$[z^m]\left((C_{\mathbf N}z-1)^n{\bf q}_{\bf N}(z)\right)$$

is positive for every $m\ge n+1$. This would extend the theorem discussed immediately before the conjecture and is proposed as an alternative expression useful for studying fine analytic properties of ${\bf q}_{\bf N}(z)$ and ${\bf z}_{\bf N}(q)$.