Period-doubling symbol-sequence conjecture for quadratic diffeomorphisms

Consider the symbol sequences of periodic orbits of the map in the paper, with a period-doubled orbit obtained from a parent periodic orbit. When the sequences are ordered appropriately, doubling the parent sequence means repeating its symbols twice. Period-doubling sequence conjecture. Symbol sequences between period-doubled periodic orbits have exactly one differing symbol; equivalently, the sequence of the doubled orbit is obtained by doubling the original sequence and flipping the first sign. This pattern was observed for finitely many period doublings and is related to the corresponding pattern for one-dimensional maps, but it remains unproved in the stated setting.