The answer is no.

Take two parallel circles of unit radius in $z=\pm \epsilon$ with $\epsilon$ small. Tilt the two circles very slightly toward one another. This satisfies your hypotheses. There are clearly at least three minimal surfaces with this boundary the tilted flat disks and a stable and unstable annulus (obtained by perturbing the appropriate pieces of a catenoid).

If you want non-uniqueness in the class of disks then form a small bridge between the two closest points on the circle. This boundary also satisfies your hypotheses. The bridge principle implies there is a stable minimal surfaces with this boundary that should look like the two flat disks joined by a small bridge. However, it is pretty clear the area minimizing disk is the "ribbon" solution going around the outside. Hence, you have two distinct minimal disks.