Assistant Chief Fire Officer Simon Moore said Mr Cox was not on a call at the time.
He added: "The circumstances of his death have not been established. We are saddened and upset by this tragic news and our thoughts are with his family and colleagues at this very difficult time."
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The entertainer was held for the murder of Stuart Lubbock, who was found dead in his swimming pool in 2001, but never charged.
Although the diamond was believed to be cut in the 19th Century, its history and how it arrived at the car boot sale is unknown.