If $f(n)=\omega(h(n))$ and $g(n)=o(h(n))$ then is $f(n)=\Theta(g(n))$?

Question

My question is exactly what the title says. If I have that $f(n)=\omega(h(n))$ and $g(n)=o(h(n))$ hold, then does $f(n)=\Theta(g(n))$ hold as well? My intuition says that the second part is false, but I couldn't put this idea on paper.