The finite-space characterization of Lebesgue covering dimension

Let $\partial\Delta^n$ be a finite topological space weakly homotopy equivalent to the sphere $\mathbb S^n$, with $\mathbb S^n\to\partial\Delta^n$ a trivial Serre fibration, and let $\check{\mathrm{H}}_q(X)$ denote the chosen Čech $q$-th homology group. Let $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$ be the indicated finite-space maps. The Lebesgue-dimension conjecture. For a finite CW complex $X$,

$$\check{\mathrm{H}}_q(X)=0\quad\text{for }q>n$$

if and only if

$$X\to\{\bullet\}\in\{\partial\Delta^n\to\{\bullet\},\,\mathsf{W}\to\mathsf{V},\,\mathsf{V}\to\{\bullet\}\}^{lr}.$$

The source explicitly leaves the homological theory underspecified, so the proposed equivalence is open and tentative.