I would go along with the advice on increasing the natural light. From the images it is evident that low light is an issue. I can't be specific regarding the wilting but in part that can also be a symptom of lack of light.

Two things to always remember regarding the Rex Cultorum is that you must never plant the rhizome low; it must always sit on the soil surface. Also, don't allow the foliage to remain wet. I would never syringe or water these begonias overhead. Whilst they should not reach wilting point due to lack of water, be careful not to over water especially if the medium is over retentive of water and may become stagnant.

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