Conjecture on optimal responses in the flipped Penney-Ante game

Assume Player I chooses a binary string $A=a_1\dots a_n$. In the flipped Penney-Ante game, the player whose string appears last wins. A response string for Player II is optimal if it maximizes Player II's winning probability against $A$.

Flipped-game optimal-response conjecture. The best response strings for Player II are one or more of

$$HH\dots H,\quad TT\dots T,\quad a_2\dots a_nH,\quad a_2\dots a_nT.$$

The conjecture is motivated by computations through $n=10$ and by the displayed examples, where optimal responses follow this four-form pattern. The source does not provide a resolution, and it remains open.