Letters to the Editor
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Maybe I'm a wuss
But I'd just go with the liability/insurance excuse. Why provoke a confrontation with a relative when it's the husband, Mr. "I Wanna Be a Bigshot Pool-Owning Neighborhood Hero," who's caused the whole problem in the first place?
Waiting for hubby to address it will do no good, as he clearly lacks any understanding as to why this might be a problem in the first place.
Lengthy, apologetic, long-winded excuses will come across as forced and insincere; in addition, there's a very good possibility, if the cousin is clueless or evil enough, she'll use any apologetic excuse as a point of negotiation and try to bargain her way back to unfettered pool access. I've had conversations like that; they're exhausting and unproductive.
A blunt "get the hell out" might make LW feel better momentarily, but it will also likely embarrass and/or anger the cousin, who will spend the next 20 years making obnoxious, sarcastic comments at every family gathering and badmouth LW to all the other relatives.
Nope. I vote for the liability option. It's perfectly rational, justifiable, and finite - there's no arguing with an insurance company.

