Yes, sort of, I found something similar in regards to base paths. This was in response to a question of if someone wanted a triple instead of a HR to complete a cycle, could they stop at third and record the triple instead of the HR.
“If I remember correctly, on May 26, 1959 in a game at County Stadium in Milwaukee pitting the visiting Pittsburgh Pirates against the Milwaukee Braves, this exact issue became a reality. Pitcher Harvey Haddix for Pittsburgh was outstanding. For 12 consecutive innings the Braves went 3 up, 3 down, for 36 consecutive outs. In the bottom of the 13th inning Felix Millan reached first base on an error by Don Hoak, Henry Aaron walked, and then Joe Adcock blasted what everyone thought was a 3-run walk off home run. However, after passing second base and watching Felix Millan score the winning by run, Henry Aaron just left the base path because the game was over. This ended up being scored as a double for Joe Adcock and he received credit for a RBI, Henry Aaron was determined to be out, and the Braves won the game 1-0.”
The rules do state that the runner must touch all bases for the run to count, and cannot pass ahead of other runners on the bases in front of them. So there is some order to hitting and scoring a HR. However, I think it would only matter in regards to base paths if it was very blatant and the opposing team challenged it with the umpires. Generally, base path rule only gets called into question when there is a tag involved, but I do think technically there is a rule and if the runner abandoned the base running, they’d be called out and his run would not count. But I highly doubt it would ever by challenged in the scenario you posted.