Background

For a topological space $X$, let $R(X)$ be the category of retractive spaces over $X$. An object of this category is a space $Y$ equipped with maps $s: X \to Y$ and $r: Y \to X$ such that $r\circ s$ is the identity. These are called structure maps. A morphism of $R(X)$ is a map of underlying spaces which commutes with the structure maps.

Let $\text{Top}$ be the category of spaces with the model structure given by weak homotopy equivalence, Serre fibration and Serre cofibration. Quillen showed that $R(X)$ inherits the structure from $\text{Top}$ where a map of $R(X)$ is a weak equivalence/fibration if and only if it is so after applying the the forgetful functor $R(X) \to \text{Top}$.

In this model structure one can characterize a cofibration $A \to Y$ as follows:

(i) up to isomorphism, $Y$ is obtained from $A$ by sequentially attaching cells, where a cell is an object of the form $X \amalg D^n$ and the attaching is done with respect to a morphism from $X \amalg S^{n-1}$. In this way the pair $(Y,A)$ is a relative cell complex over $X$.

(ii) More generally, we must also allow retracts of the pairs $(Y,A)$ appearing in (i).

Observation

If $f: X \to X'$ is a map, then pullback along $f$ defines a functor $$ f^* : R(X') \to R(X) $$ which fails to be exact. Even if $f$ is a fibration this functor can fail to be exact since it need not preserve cofibrations. For example if $X' = *$ is a point, and $Y$ a based cell complex (which is a cofibrant object of $R(*)$), then $f^*(Y)$ has underlying space $X \times Y$, and although I don't at the moment have an easy explanation as to why this needn't be cofibrant, I am pretty sure it isn't in general.

Questions

I am looking for an alternative model structure on $R(X)$, having the same weak equivalences (= weak homotopy equivalence) as above, but which has the property that base change along fibrations preserves cofibrations. I have a candidate for what the cofibrations should be. I'd like to know if it can be completed to a model structure.

A cofibration $A \to Y$ in this conjectured model structure is an inclusion which is up to isomorphism gotten by starting with $A$ and attaching "cells," but now a cell is something of the form $X\times D^n$ and we are doing the attaching along a morphism out of $X\times S^{n-1}$. More generally, we need to allow retracts of these kinds of inclusions $A \to Y$.

Question 1: If we define the fibrations by the lifting property with respect to the trivial cofibrations, does this form a model structure on $R(X)$?

Question 2: If the answer to Question 1 is yes, then can we explicitly pin down the fibrations?

Remark

In the model structure defined by Quillen, the quotient space (= cofiber) $Y/s(X)$ arising from a cofibrant object $Y$ of $R(X)$ is a cofibrant object of $\text{Top}$.

In the model structure I'm proposing, if $Y$ is cofibrant, then the fibers of the structure map $r: Y \to X$ will be cofibrant spaces