The finite-space characterization of fibrations

Let $f:X\to B$ be a cellular map of finite CW complexes. Let $P$ be the finite-space map from the three-point diagram with one distinguished point and two indistinguishable points to its quotient identifying the distinguished point with that pair, and let $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$ be the indicated maps. The finite-space fibration conjecture. The map $f$ is a fibration if and only if it factors as $f=f_lf_{lr}$ with

$$f_l\in P^l,\qquad f_{lr}\in P^{lr}\cap\{\mathsf{W}\to\mathsf{V},\,\mathsf{V}\to\{\bullet\}\}^{lr}.$$

Moreover, every locally constant map over a paracompact space, and more generally every numerable fibre bundle of separable metric spaces, admits such a decomposition. The source presents this as a conjectural finite-space description, with attribution to Dold for the numerable-bundle context.