Relation between the two probability densities

Suppose $X_1, X_2, Y_1, Y_2$ are independent random variables on the same probability space with densities $f_1,f_2,g_1,g_2$ respectively.

If $$\int_{x} f_1(x)f_2(z-x) \,dx = \int_{y} g_1(y)g_2(z-y) \,dy \quad (*)$$ for all feasible $z$ and $$\frac{f_i(x)}{g_i(x)}$$ is non-decreasing in $x$ for all $x$ in the support of $X_i$ and $Y_i$ for both $i\in\{1,2\}$

then can we say something about the relationship between $f_i$ and $g_i$ for both $i\in\{1,2\}$?

Thanks in advance for any kind of help.

The support of the random variables $X_1, X_2, Y_1, Y_2$ is $[0,a]$ for some $a>0$.
Also I know that $f_i(x)=g_i(a-x)$ for all $x$ and for both $i\in\{1,2\}$.
I concluded that $g_i(0) = g_i(a)$ using $(*)$ with $z=2a$ and hence $f_i(0) = f_i(a)$. I think this is correct, just wanted to scrutinise this statement over stackexchange.