A parabolic minimal surface in R3 intersects every nonflat properly embedded minimal surface of bounded curvature

We show that the image of a nonconstant conformal harmonic map $\mathbb{C}\to {\mathbb{R}}^3$, not necessarily proper and possibly with branch points, intersects every properly embedded nonflat minimal surface of bounded curvature in ${\mathbb{R}}^3$. The same holds if $\mathbb{C}$ is replaced by any open conformal surface without nonconstant bounded subharmonic functions.