The current English translation is very awkward. Something like "Are you starting to wash up?" or "Are you starting to wash the dishes?" would be more natural, and a more consistent use of mosogatni based on previous examples' uses.
It implies that, well, it has been started. Something happened in the past, sometime. The Hungarian sentence doesn't suggest anything about the past, it's about right now. Your sentence would be like "Elkezdtél (már) mosogatni?"