The dimension-function conjecture for restrained structures

Let $\mathbb K$ be restrained. A dimension function is a function $d$ assigning to every definable set a number in $\mathbb N\cup\{-\infty\}$ and satisfying the following axioms for definable sets $A,B\subseteq\mathbb K^n$ and $C\subseteq\mathbb K^{n+1}$: $d(A)=-\infty$ if and only if $A=\emptyset$, $d(\{a\})=0$ for each $a\in\mathbb K$, and $d(\mathbb K)=1$; $d(A\cup B)=\max(d(A),d(B))$; $d(A^\sigma)=d(A)$ for every permutation $\sigma$ of $\{1,\dots,n\}$; and, for $C(i)=\{\bar x\in\mathbb K^n:d(C_{\bar x})=i\}$ with $i=0,1$, each $C(i)$ is definable and $d\bigl(C\cap(C(i)\times\mathbb K)\bigr)=d(C(i))+i$. The dimension-function conjecture. There is such a dimension function $d$ that coincides with $\dim$ on $\mathcal D_\Sigma$ sets. The source notes that the conjecture is known in several cases, but does not state a general resolution.