Let $V=\bigoplus_{d\in\mathbb N}V(d)$ be a Möbius-covariant vertex algebra with $V(0)=\mathbb C$.

Recall that a vector $v\in V$ is called *quasi-primary* if $L_1v=0$.

For $v\in V(d)$, we write $Y(v,z)=\sum_{n\in\mathbb Z} z^{-n-d}v_{(n)}$.

With that convention, $v_{(n)}$ is an operator $V(k)\to V(k-n)$.

Let $d>n$.

Is it true that for any quasi-primary $v\in V(d)$ and any vector $w\in V(n)$, we have $v_{(n)}w=0$?

If the above relation does not always hold, are there reasonable extra assumptions that one can impose on $V$ that imply it?

Do the above relations hold when $v$ is required to be primary instead of quasi-primary?

(Add the assumption that $V$ is a VOA so that the notion of a primary vector make sense)