Folklore conjecture on the fixed points of the Minkowski question mark function

For $x\in[0,1]$, let $?(x)$ denote the Minkowski question mark function, defined for an irrational continued fraction $x=[a_1,a_2,\ldots]$ by

$$?(x)=\sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{2^{a_1+\cdots+a_k-1}},$$

and for rational $x$ by the corresponding finite sum. Folklore conjecture. The function $?(x)$ has exactly five fixed points. Moreover, there is exactly one irrational fixed point of $?(x)$ in the interval $(0,\frac{1}{2})$. The existence of fixed points on either side of $\frac12$ follows from continuity and the values $?(0)=0$, $?(\frac12)=\frac12$, and $?(1)=1$; the conjecture asserts the total number of fixed points and the uniqueness of the irrational fixed point in the left interval.