Let $X$ be a smooth, projective variety, and $E_1$ and $E_2$ be two smooth divisors on it meeting each other normally. Suppose that $D^b_{E_1}(X) \subset D^b(X)$ and $D^b_{E_2}(X) \subset D^b(X)$ are admissible subcategories, so that there are semiorthogonal decompositions $$ D^b(X)=\langle D^b_{E_1}(X)^{\perp}, D^b_{E_1}(X) \rangle $$ and $$ D^b(X)=\langle D^b_{E_2}(X)^{\perp}, D^b_{E_2}(X) \rangle $$.

Is then $D^b_{E_1\cap E_2}(X) \subset D^b_{E_i}(X)$ admissible?

Or is there any further condition that can guarantee this?