Uniqueness conjecture for continuous solutions of the Furstenberg function equations

Let $p,q>1$ be integers, and let $C[0,1]$ denote the continuous functions on $[0,1]$. Consider functions $f\in C[0,1]$ satisfying that $f$ is non-decreasing and

$$f(x)=\sum_{i=0}^{p-1} f\left(\frac{x+i}{p}\right)-f\left(\frac{i}{p}\right)=\sum_{i=0}^{q-1} f\left(\frac{x+i}{q}\right)-f\left(\frac{i}{q}\right).$$

The function-equation conjecture. The only such function is $f(x)=x$. The conjecture is presented as a sufficient condition for Furstenberg's conjecture on continuous ergodic $\times p,\times q$-invariant measures: distribution functions of continuous invariant measures satisfy these equations. The parenthetical discussion notes that, using Furstenberg's classification of closed $\times p,\times q$-invariant subsets, one may assume $f$ is strictly increasing and hence a homeomorphism of $[0,1]$ with $f(0)=0$; the supplied text gives no resolution.