The dimension inequality for definable images of $\mathcal D_\Sigma$ sets

Let $C\subseteq\mathbb K^m$ be a $\mathcal D_\Sigma$ set, and let $f:C\to\mathbb K^n$ be definable, without assuming that $f$ is continuous. The dimension-image conjecture. One has

$$\dim(f(C))\leq\dim(C).$$

In particular, there is no surjective definable function from $\mathbb K^n$ to $\mathbb K^{n+1}$. This conjecture is presented as implied by the dimension-function conjecture; the source gives only a partial result, so the general claim remains open.