The un-named powers that be have told the press that a new Cabinet and a new LDP leadership lineup will be announced on November 2. When asked for the reason why the change would happen on that date, the source(s) explained that the special session of the Diet ends on November 1.
The PM has already indicated that he will offer his political support to the prime ministerial candidate who turns in the best performance over the next 11 months. Given the importance of securing an influential post in this next round, it is not unreasonable that the PM and his people are trying to minimize the amount of time the candidates have for public displays of jockeying for position.
Coming after a general election, the changes in the next Cabinet are likely to be far more dramatic than a simple reshuffle. I have already made some rather speculative guesses here about some of the main Cabinet and LDP leadership positions.