Map the function $Y\to *$ into the left edge to get two homotopy pushouts in a row; this would produce a cofiber sequence $Y\to X_n\to X_{n+r}$. But this can't be done in general.

Map the function $Y\to *$ into the left edge to get two homotopy pushouts in a row; this would produce a cofiber sequence $Y\to X_n\to X_{n+r}$. But this can't be done in general.

One interesting way to see that this is not the case in general is that it would lead to cone decompositions of a generic space with length roughly $\log_2$ of the dimension. This is impossible for spaces with high ratio of cone length to dimension, such as projective spaces.