Start by choosing two of the points $A$ and $B$ such that all other points lie on the same side of the line $AB$. (This can be done by moving a line from the distance to the points until it meets a point, and then turning the line until you meet another point.) It there are other points on the line AB, you should choose $A$ and $B$ such that the other points lie between $A$ and $B$.
Now consider circles through these two points with various radius where the larger part of the circle lies on the side of the other points.
Then a circle with larger radius covers all the area of a circle of smaller radius (inside the half space where the points lie). Hence, the biggest circle you need to cover $A$, $B$ and a third point already covers all points.
So, $A$, $B$ and the third point for which one needed this biggest circle are the three points in your conjecture.