MacPherson's conjecture on omitting the dimension axiom

Let $X$ be a stratified space with stratification ${\cal S}$, let ${\cal P}(X,{\cal S})$ denote MacPherson's category of perverse sheaves, and let $D^b_{\cal S}(X)$ denote the bounded derived category of ${\cal S}$-constructible sheaves. Consider the category obtained from MacPherson's definition by omitting its dimension axiom. MacPherson's conjecture. This resulting category is equivalent to

$$D^b_{\cal S}(X).$$

The conjecture concerns the equivalence between MacPherson's axiomatic construction and the constructible derived category. The surrounding text states that agreement with the original Beilinson–Bernstein–Deligne definition is a goal of the paper, but supplies no proof or resolution of this conjecture.