Assume a minimal surface $\Sigma$ has boundary on the unit sphere in the Euclidean space
and $r$ is the distance from $\Sigma$ to the center of the ball.
Is it true that 
$$\mathop{\rm area} \Sigma\ge \pi\cdot(1-r^2).$$


Comments:
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* The problem is solved in all dimensions and codimension, see ["Area bounds for minimal..."](http://arxiv.org/abs/1607.04631) by Brendle and Hung in 2016. (Thanks Rbega for the ref.)
* If $r=0$, the statement follows directly from the monotonicity formula.
* If $\Sigma$ is **topological disc** the answer is YES, see answer of Oleg Eroshkin below.
* The general question is formulated as a conjecture in 1975 --- see comment of Ian Agol.
* There is an analog in all dimension and codimension for **area minimizing** surfaces, see Alexander, H.; Hoffman, D.; Osserman, R. *Area estimates for submanifolds of Euclidean space.* 1974.



Minimal surface in a ball
This has just been [solved](http://arxiv.org/abs/1607.04631) (in full generality) by Brendle and Hung using the first variation formula together with a clever (if mysterious) choice of vector field.


