It seems to me that there an obvious Whitehead-manifold inspired construction to try here to produce a counterexample. Let w : S^1 x D^2 -> S^1 x int D^2 be the familiar Whitehead embedding, which is null-homotopic but not “null-isotopic”, for which the two inclusions S^1 x bdy D^2  \subsetinclusion  S^1 x D^2 \setminus  w(S^1 x int D^2) and w(S^1 x bdy D^2)  \subsetinclusion  S^1 x D^2 \setminus  w(S^1 x int D^2) are each pi_1-injective (google Whitehead manifold.) For i = 1 and 2 let w_i : T^2 x D^2 -> T^2 X D^2 be the two obvious product embeddings engendered by w, namely, let w_2 : = iden(S^1) x w : S^1 x S^1 x D^2  -> S^1 x S^1 x int D^2, and similarly let w_1 = “w x iden(S^1)” (the quotes meaning that you do the appropriate factor permutations here to make it work). It looks as though the contractible direct-limit 4-manifold M^4 gotten from the sequence w^1, w^2, w^1, w^2, … is not simply-connected at infinity (like the Whitehead 3-manifold). True?