If $f : \mathbb{R}^n \to \Bbb R$ is monotone over all lines, can it be written as $h \circ l$ where $h$ is monotone and $l$ is a linear form?

Let be $f : \Bbb R^n \to \Bbb R$ monotone over all lines (not affine ones, but if there is an answer over affine lines, I'm interested.)

Is it possible to find $h : \mathbb{R} \to \Bbb R$ monotone and $l : \Bbb R^n \to \Bbb R$ linear so that $f = h \circ l$ ?

I tried to look by supposing I have such a factorization, and as $\ker l$ is a hyperplane, I have $n - 1$ lines where $f$ is constant. I tried to use the monotonicity condition by trying to compare $f(0)$ and over lines, but it didn't work.

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