Estimate, a racehorse owned by the Queen and trained by Sir Michael Stoute, has tested positive for morphine, a prohibited substance, Buckingham Palace announced.
Initial indications are that the positive test resulted from the consumption of a contaminated feed product, the palace said.
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Duncan Jones tweeted: "1 month since dad died today. Made this card for him at Christmas. Due in June. Circle of life. Love you, granddad."
More than 10 days after his body was found, Richard Cole's family claim they are still in the dark over the circumstances of his death.
A cold and frosty start, then staying largely dry with plenty of sunshine.