The model-structure conjecture generated by finite topological spaces

Let $\mathsf{W}\to\mathsf{V}$ and $\mathsf{V}\to\{\bullet\}$ be the indicated maps of finite topological spaces, and let $\operatorname{Top}$ denote the category of topological spaces. The finite-space model-structure conjecture. A cellular map of finite CW complexes is a trivial fibration if and only if it lies in

$$\{\mathsf{W}\to\mathsf{V},\,\mathsf{V}\to\{\bullet\}\}^{lr}.$$

Moreover, every locally trivial fibre bundle with contractible fibre over a paracompact space, and more generally every numerable fibre bundle with contractible fibre whose total space is separable metric, lies in this class; and there is a model structure on $\operatorname{Top}$ for which this class is the class of trivial fibrations. The source gives evidence and related speculations but no proof of the full assertion.