If I understand correctly, the claim below follows from some well-known facts about the Quillen and Hurewicz model structures on the category of all topological spaces:
If $p : X \to Y$ is a Hurewicz fibration and (half of) a homotopy equivalence, then $p$ has the right lifting property with respect to the boundary inclusions $\partial B^n \hookrightarrow B^n$ for all $n \ge 0$.
Question. Is there a direct proof using the homotopy lifting property?
The case where $p : X \to Y$ is (isomorphic to) a product projection $Y \times F \to Y$ where $F$ is contractible is straightforward enough. The general case seems to boil down to the following fact:
There exists a map $s : Y \to X$ such that $p \circ s = \mathrm{id}_Y$ and a homotopy $\alpha : X \times [0, 1] \to X$ such that $\alpha (x, 0) = x$, $\alpha (x, 1) = s(p(x))$, and $p(\alpha(x, t)) = p(x)$.
This is Exercise 15 in §4.3 of [Hatcher, Algebraic topology], but the proof of this appears to require some technical point-set topology which I would rather avoid...