Hurewicz fibration conjecture for randomized simplicial sets

Let $F$ be a simplicial set, with geometric realization $|F|$, and let $L(F)$ denote its realization in the deterministic construction. Let $\mathbb{L}(F)$ be the randomized realization of the underlying pre-simplicial set, with probability-law map to the ordinary realization $\|F\|$.

Hurewicz fibration conjecture. The maps

$$L(F) \to |F| \qquad\text{and}\qquad \mathbb{L}(F) \to \|F\|$$

are Hurewicz fibrations.

This conjecture proposes a fibration enhancement of the comparison maps between the paper's randomized and ordinary realizations. The preceding results establish homotopy-equivalence properties for the corresponding maps in the pre-simplicial setting, but do not establish the stronger Hurewicz fibration assertion.