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)