Let $M$ and $N$ be compact $n$-dimensional manifolds with a common (nicely embedded) compact submanifold $S$ (we may assume that the inclusion of $S$ in $M$ and $N$ is $\pi_1$-injective). Let $M\#_S N$ denote the connected sum of $M$ and $N$ along $S$ (see Wikipedia). I wish to study necessary conditions for $M\#_S N$ to be aspherical!

In the case of usual connected sum $M \#N$, we have that $M \# N$ is **not** aspherical if $M$ is not homotopic to $S^n$. In the above general connected sum $M\#_S N$, we have that $M\#_S N$ is **not** aspherical if $\pi_1(M\#_S N)$ is an amalgamated free product or an HNN extension over a subgroup of cohomological dimension less than $n-1$. But what happens if $\pi_1(M\#_S N)$ is neither? Are there any other sufficient conditions for $M\#_S N$ to be **not** aspherical?

I am considering the following simple example in dimension $4$. Let $M = S^2 \times T^2$ and $N = T^4$, where $T^n$ denotes the $n$-torus. Let $S = T^2$ and form $M\#_S N$ with $\pi_1$-injective inclusions. I think $\pi_1(M\#_S N) = \mathbb{Z}^4$ (the fundamental group of the $4$-torus). How do I decide whether $M\#_S N$ is aspherical in my case? If $M\#_S N$ were aspherical, then $M\#_S N = T^4$. Can I eliminate this possibility?