You can say it is not true, but that doesn't make it so. The truth is unclear. Truman felt hitting one city (Hiroshima) was enough. One U.S. position was considering forcing the Japanese high command to surrender by warning them we would not stop the Soviets from invading, and allowing the Soviets to do so. The Japanese knew the Soviets would be brutal. Another U.S. position (our military command) was that we block the Soviets by a second nuclear strike to end the war. Truman didn't want the second nuke dropped. Unknown to him, the second bomb was pre-authorized for a drop as soon as it was ready, without having been tested. There were many (including Truman) who felt the Japanese would have surrendered just at the threat of unleashing the Soviets on them. Our military wanted the second bomb dropped to both force Japanese surrender and to keep the Soviets out of Japan.
So it's not just the Japanese Emperor not having control of the Japanese military, but our civilian leadership (White House) was straining to control our military leaders. Perhaps this helped lead to Truman firing MacArthur in 1951. (I feel MacArthur made the right choices, not Truman.)
Do you really think that if Japan had surrendered after Hiroshima that the USA would have dropped a second bomb? Nope.
As I said, MacArthur was the main reason were able to keep Russia out of Japan. He refused to allow it, unlike FDR and Churchill did in Europe.