Zudilin's integrality conjecture for generalized canonical coordinates

Let $N_1,N_2,\dots,N_k$ be positive integers, and let $\mathbf{N}=(N_1,N_2,\dots,N_k)$. Let $\mathbf{F}_{\mathbf{N}}(z)$ and $\mathbf{G}_{\mathbf{N}}(z)$ denote the generalized power series used to define the canonical coordinate

$$\mathbf{q}_{\mathbf{N}}(z):=z\exp\!\left(\frac{\mathbf{G}_{\mathbf{N}}(z)}{\mathbf{F}_{\mathbf{N}}(z)}\right).$$

Zudilin's conjecture. For any positive integers $N_1,N_2,\dots,N_k$,

$$\mathbf{q}_{\mathbf{N}}(z)\in z\mathbb{Z}[[z]].$$

This is the formulation attributed to Zudilin in the source; the source says it is equivalent to Zudilin's original formulation. The supplied text gives no resolution status.