Attempt at proving a proposition about the conjugate of the Wigner Transform,where is my mistake?$\overline{W(f,g)}=W(g,f)?$

Question

Edit,this is the exact phrasing of the question

I define the Wigner Transform as follows,it is the fourier transform of the fourier-wigner transform Here is the question Here is the exact phrasing of the question For all Swartz Functions $f$ and $g$ on $\mathbb{R^n}$

$W(f,g)(x,z)=(2\pi)^{-n/2}\int_{\mathbb{R^n}}e^{-iz.p}f(x+p/2)\overline{g(x-p/2)}dp$.where $x,z\in\mathbb{R^n}$

I want to show that $\overline{W(f,g)}=W(g,f)?$ Here is my Attempt at the question

Here is my attempt at the question Could someone point out my mistake please?I hope my pictures are readable?

I was thinking about using the adjoint formula for the fourier transform but the bar is for conjugate,not the fourier transform?