Here is a bit of Mathematica code that rather supports Joseph's conclusion.
Tet[phi_] := {{Sin[phi], 0, Cos[phi]}, {-Sin[phi], 0, Cos[phi]}, {0, Sin[phi], -Cos[phi]}, {0, -Sin[phi], -Cos[phi]}};
Rect[a_, phi_] := Module[{x, y, u, w}, {x, y, u, w} = Tet[phi]; Polygon[{a x + (1 - a) u, a x + (1 - a) w, a y + (1 - a) w, a y + (1 - a) u}]];
v = Subsets[Range[4], {3}];
Manipulate[ Graphics3D[{Opacity[0.2], Sphere[{0, 0, 0}, 1], Opacity[0.3], GraphicsComplex[Tet[phi], Polygon[v]], Opacity[0.8], Rect[a, phi]}], {phi, 0, Pi/2}, {a, 0, 1}]
Using parameters like a=0.9, phi=1.4 one obtains an elongated rectangle inscribed a flat tetrahedron, close to an equatorial plane. The maximal inscribed ellipse in this rectangle hardly is contained in any triangle that fits in the unit ball.