There will never be a Referendum on dissolving "parts of the Monarchy", there might conceivably be a referendum on abolition but that is highly unlikely. It is far more Likely the monarch would see the writing on the wall and voluntarily abolish their position and usher in a Republic.
After all, the Monarch has been placed over us by God to Protect us. I'm not joking here, this is widely acknowledged to be the view of HM Queen. Abolition in the UK is unlikely though, the Queen is outstandingly popular, her son is also generally very popular and her grandson is possibly the post popular of the lot.
On the other hand, Abolition in her other Dominions is a real possibility, but it all depends on timing. The Australians have already rejected abolition under the current Monarch. It's possible that once Charles ascends abolition there and in Canada will move forward but not immediately after his coronation - it would take a few years at least to get going.
In that time he might prove an excellent King, and if he dies early and William ascends then abolitionists will be back to square one, I think.
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