Assume one is given a commutative square of spaces

$A \quad \to \quad C$

$ \downarrow \qquad \qquad \downarrow$

$B\quad \to \quad X$

which is a pushout and in which each map is a cofibration. If $A \to B$ is $r$-connected and $A\to C$ is $s$-connected, then the Blakers-Massey theorem says that the square is $(r+s-1)$-cartesian (this means that the map from $A$ into the homotopy pullback of the remaining terms is $(r+s-1)$-connected).

The only proofs of this statement at this level of generality that I know of make use of transversality. However, if all spaces are simply connected, there are proofs which avoid transversality (for example, when $B$ is a point, one can deduce it using the Serre exact sequence).

Question: Is transversality intrinsic to a proof of the theorem in the general case?