Let $\Sigma_1$ and $\Sigma_2$ be two properly embedded complete connected minimal surfaces (without boundary) in $\mathbb{R}^3$. Let $p \in \Sigma_1 \cap \Sigma_2 \ne \emptyset $.
It is well known that if $\Sigma_1$ lies on one side of $\Sigma_2$, then $\Sigma_1 = \Sigma_2$. (See for instance Corollary 1.28 of A Course in Minimal Surfaces, by Colding & Minicozzi).
Can I have the same conclusion assuming that at the contact point $p$ the surfaces agree up to the second order (without assuming that one lies above the other)?
Any help will be very apreciated. Thanks!