Just some remarks:

0) If the theorem is true, then it is tight in the following example:

Draw a circle A, that represents a slice of the ball.

Inscribe inside A a square B, which represents a very thin tetrahedron.

Inscribe inside B a square C joining the midpoints of the sides of B. This square represents the midbase of the tetrahedron.

Inscribe inside C a circle D, which is the ellipsoid.

In this case we have concentric circles with radius(A)=2radius(D), whicch is just enough to fit a triangle between A and D.

This example makes the problem beautiful for me.

1) If the theorem is true, then it is true also for the case in which the outer ball is generalised to an ellipsoid. Why? because by changing the inner product of the space, the ellipsoid turns into a ball. 

2) So the context of our problem is affine geometry of $\mathbb R^3$ (that is, we can drop the inner product). In fact, we can drop even the affine structure and keep only the projective structure.

3) We can then settle a new inner product so that the ellipse is a circle.