Cofibration conjecture for closed subspaces of reasonable and PA-spaces

Let $X$ be a reasonable topological space or a PA-space, and let $Y\subset X$ be a closed subspace. A map is a cofibration if it has the homotopy extension property. Cofibration conjecture. The natural inclusion

$$Y\hookrightarrow X$$

is a cofibration in both the category of reasonable topological spaces and the category of PA-spaces. This would extend the preceding compact-subspace cofibration result from compact subspaces to arbitrary closed subspaces; the supplied text gives no resolution of the claim.