Are there two distributions which maximize $\int g_0^ug_1^{1-u}d\mu$ but do not minimize all f-divergences?

Question

Here is the question:

Are there two sets of density functions, say $\mathcal{G}_0$ and $\mathcal{G}_1$, for which the two densities $\hat g_0\in \mathcal{G}_0$ and $\hat g_1\in \mathcal{G}_1$ maximize $$\int g_0^u g_1^{1-u} d\mu$$ for all $u\in(0,1)$ but the same pair of densities $(\hat g_0,\hat g_1)$ dooes not minimize $$\int \phi\left(\frac{g_0}{g_1}\right)g_1 d\mu$$ for all convex $\phi$ with $\phi(1)=0$?

I have the feeling that there should be such an example but I failed to find one. Any ideas?