Strict containment conjecture for weakly locally finite and Stewart's division rings

Let a weakly locally finite division ring be a division ring in which every finite subset generates a centrally finite division subring over the prime subfield, and let a Stewart's division ring be a division ring that is locally finite over some subfield. The class of Stewart's division rings is contained in the class of weakly locally finite division rings.

Strict containment conjecture. The class of weakly locally finite division rings strictly contains the class of Stewart's division rings.

The preceding theorem establishes the containment, so the conjecture asserts that the containment is proper. Equivalently, it predicts the existence of a weakly locally finite division ring that is not a Stewart's division ring; the supplied text does not state whether such an example is known.