Dog Dies After Being Shot

7:22 AM PT -- Hope says Conan died Saturday night ... "We're brokenhearted to share that Conan passed away from blood loss last night. He fought up until the very end. We're crushed," adding, "Just a dog running through the woods, trying to make his way home. We bought 60 acres so our dogs have space to run and live their most full lives. Conan was kind and loving." And then she said, "As animal lovers we are struggling to make sense of anyone using their right to own guns to shoot pets of any kind."

Hope Solo claims some gun-toting maniac shot one of her dogs Wednesday night -- but the good news, the pooch survived the attack.

It all went down North Carolina, where the former Team USA soccer…