  For all open Riemann surface M and real number $\theta \in (0,\pi/4),$ we
construct a conformal minimal immersion $X=(X_1,X_2,X_3):M \to \mathbb{R}^3$
such that $X_3+\tan(\theta) |X_1|:M \to \mathbb{R}$ is positive and proper.
Furthermore, $X$ can be chosen with arbitrarily prescribed flux map.
  Moreover, we produce properly immersed hyperbolic minimal surfaces with non
empty boundary in $\mathbb{R}^3$ lying above a negative sublinear graph.
