Assume that $V$ is a vertex operator algebra, and the VOA $V'$ is a vertex subalgebra of $V$. The notion that $V'\subset V$ is a conformal inclusion has different meanings in different literatures. Physicists often mean that $V'$ and $V$ have the same central charge, while mathematicians often mean that $V'$ and $V$ have the same conformal vector. Clearly the second meaning implies the first one. Is the converse also true? i.e., if $V'$ and $V$ have the same central charge, is the conformal vector of $V'$ also a conformal vector of $V$? (Feel free to add conditions on $V$, e.g., CFT type, rational, etc.)