Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following conjecture true?

For every Lebesgue null set $N \subseteq I\setminus S$, the preimage $f^{-1}(N)$ is again a Lebesgue null set.

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following conjecture true?

For every Lebesgue null set $N \subseteq I\setminus S$, the preimage $f^{-1}(N)$ is again a Lebesgue null set.

Edit: As pointed out by Christian Remling in the comments, it seems that this is false in general. Are there assumptions on $f$ that guarantee that the conjecture holds? I am particularly interested in whether absolute continuity of $f$ is sufficient.

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monontonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following conjecture true?

For every Lebesgue null set $N \subseteq I\setminus S$, the preimage $f^{-1}(N)$ is again a Lebesgue null set.

Let $I\subseteq \mathbb{R}$ be a closed bounded interval and $f:I \to I$ a monotonic increasing function and $S$ the countable set of points $s$ such that $|f^{-1}(s)| > 1$. Is the following conjecture true?

For every Lebesgue null set $N \subseteq I\setminus S$, the preimage $f^{-1}(N)$ is again a Lebesgue null set.