Mappings from open Riemann surfaces to the twice punctured complex plane and the restricted validity of Gromov's Oka principle

According to Gromov the Oka principle holds for holomorphic mappings from a complex manifold $X$ to a complex manifold $Y$ if each continuous mapping $X\to Y$ is homotopic to a holomorphic mapping. Giving sufficient conditions on the target $Y$ for the validity of the Oka principle for holomorphic mappings from any Stein manifold to $Y$, he initiated a line of interesting and fruitful research. On the other hand he mentions mappings from annuli to the twice punctured complex plane as simplest example for which this Oka principle fails and draws attention to the fact that mappings from annuli play a crucial role for understanding the "rigidity" of the target $Y$ in case the Oka principle fails for mappings from some Stein manifolds to $Y$.
We will say that a continuous mapping $f$ from a finite open Riemann surface $X$ to the twice punctured complex plane has the Gromov-Oka property if for each orientation preserving homeomorphism $\omega:X\to \omega(X)$ onto a Riemann surface $\omega(X)$ with only thick ends the mapping $f\circ \omega^{-1}$ is homotopic to a holomorphic mapping. For finite open Riemann surfaces we show the existence of finitely many embedded annuli in $X$, such that $f$ has the Gromov-Oka property iff its restriction to each of the annuli has this property, and describe all mappings with the Gromov-Oka property. On the other hand we show that for $X_{\varepsilon} $ being the $\varepsilon$-neighbourhood of a skeleton of a torus with a hole, the number of irreducible holomorphic mappings $X_{\varepsilon}\to\mathbb{C}\setminus \{-1,1\}$ up to homotopy grows exponentially in $\frac{1}{\varepsilon}$.