Conjecture on the structure of extended tongues

Let $ET_\tau$ be an extended tongue of period $p>1$. Its connected components are subsets of the parameter space, and a parameter has a parabolic cycle when the corresponding map has a parabolic periodic cycle. A period doubling bifurcation occurs when an attracting cycle gives rise to an attracting cycle of twice the period.

Extended-tongue structure conjecture. The connected components of $ET_\tau$ are disjoint. The boundary of every connected component consists of two disjoint connected components: an exterior component consisting of parameters for which there is a parabolic cycle of period $p$ and multiplier $1$, and an interior component consisting of parameters for which there is a parabolic cycle of period $p$ and multiplier $-1$. Moreover, a period doubling bifurcation takes place throughout the curve of interior boundary parameters.

This conjecture extends the structure proved for the extended fixed tongue $ET_0$ to extended tongues of arbitrary period. The claim is motivated by numerical studies; the corresponding general structure is not established in the supplied text.