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My wife is happy to wash the few dishes two people generate , we really have no need for a dish washer, and I was always taught a man should only be a gentleman in the presence of ladies.
I would check with the filter manufacturer and make sure the filter is dishwasher safe. Cascade dishwashing detergent with “sheeting action” to prevent water spots might be a good idea.
Disclaimer: I don’t own a dishwasher, much prefer hand washed dishes.