Could anyone prove that this is a context free language or not?
This must be a classic in formal language theory. The Pumping Lemma will not help, because, unexpectedly, it is context-free. (I did not see the answer here, but it might be overlooked)

The $w_1$ and $w_2$ parts must have a position where they differ. Unfortunately, we cannot distinguish the middle and the two matching positions at the same time. However, we can find matching positions without knowing the middle.

With some effort you can convince yourself that your language $L$ equals
$$\{ x_1 a x_2 b x_3 \mid a,b \in \Sigma, a\neq b, |x_1|+|x_3| = |x_2| \}.$$
Once you have shown that, write a grammar using that formulation.