Function with a constant rate of change on the relative interior of a convex cone of $\mathbb R^n$ which has empty interior

Let $V$ a convex cone of $\mathbb{R}^n$ having an empty interior. Let $relint(V)$ denote its relative interior.

Let $f:V \to \mathbb{R}$. Assume that for all $v \in relint(V)$, $$\lim_{t\to0}\: \frac{f(v_1,...,v_k+t,...,v_n)-f(v_1,...,v_k,...,v_n)}{t}=a_k \in \mathbb{R},$$

that is, the rate of change of $f$ in any coordinate is constant on $relint(V)$.

Is it true that on $relint(V)$, $f$ is the restriction of an affine map $T: \mathbb{R}^n \to \mathbb{R}$ ?

I know that if $g: \Omega \to \mathbb{R}$, with $\Omega$ nonempty, open and connected in $\mathbb{R}^n$, is differentiable and has constant partial derivatives on $\Omega$, then it is indeed the restriction to $\Omega$ of an affine map (Constant derivative of a function implies is an affine map.) but the problem is here different as $V$ has an empty interior in $\mathbb{R}^n$.