I can't run the downloaded versions, so this has only been confirmed in the Supporter Web Version: Baron Oro update. Whenever I finally lose in an endless run ("Endless" being specifically any loss after beating the first loop), a new run with the same character and difficulty selection begins immediately after leaving the game over screen. I had assumed this was a feature and after playing for a while, I quit via the pause menu only to discover the new run had not saved, and "continue" was grayed out. I don't know exactly which part of this is a bug or intended behavior, but I can't imagine it's all intentional.