If $(X,A)$ has homotopy extension property (HEP), $A$ closed in $X$, then so does $(X\times I,X\times\partial I\cup A\times I)$ where $I$ is the unit interval.
Is this still true if we replace HEP by HEP with respect to a certain space? Namely if $(X,A)$ has HEP w.r.t $Y$, can we prove that $(X\times I,X\times\partial I\cup A\times I)$ has HEP w.r.t $Y$, perhaps with some extra assumptions like $X$ is a manifold and $A$ is its boundary?
P.S. I know that for paracompact Hausdorff manifold $X$ and its boundary $A$, $(X,A)$ always has HEP. However I just realized when typing this question that this can fail in non-paracompact case, for example $(O,\partial O)$, where $O=\{(x,y)\in L^{2}\mid x\geq y\}$ and $L$ denotes the closed long ray, so $\partial O$ is "the union of diagonal and $x$-axis". It can be proved that no function from $O$ to $\mathbb{R}$ vanishes exactly on $\partial O$, which by some theorem in textbook implies that $(O,\partial O)$ does not have HEP. But it does have HEP w.r.t $L$ or $I$, and it's ok for me if the lemma works in the special cases where $Y=L$ or $I$.
P.S.2 Also see this post.