Let $f,g: I \to I := [0,1]$ be continous functions satisfying $f \circ g = g \circ f$. Does there exist $x_0 \in I$ such that $f(x_0) = g(x_0)$ ?

Background: In a homework the problem was possed with $g=\operatorname{id}$ (where it can easily be solved with help of the intermediate value theorem). The lecturer said the stronger statement above is true, but he didn't knew a proof. I googled a little around, but could only find something about the "commuting function problem" (existence of a common fixed point of $f$ and $g$) that is known to be false.

Let $f,g: I \to I := [0,1]$ be continous functions satisfying $f \circ g = g \circ f$. Does there exist $x_0 \in I$ such that $f(x_0) = g(x_0)$ ?

Background: In a homework the problem was posed with $g=\operatorname{id}$ (where it can easily be solved with the help of the intermediate value theorem). The lecturer said the stronger statement above is true, but he didn't know a proof. I googled a little around, but could only find something about the "commuting function problem" (existence of a common fixed point of $f$ and $g$) that is known to be false.