Existence of neighborhood inclusion for 4-chordal graphs

Let $N(v)$ be the (open) neighbourhood set of a vertex $v$, and let $N[v]$ be the closed neighbourhood set of $v$.

A graph $G$ is called 4-chordal if $G$ has no induced cycle with five or more vertices.

Question: Does every 4-chordal graph $G$ have two vertices $x$ and $y$ such that $N(x)$ is a subset of $N(y)$ or $N[x]$ is a subset of $N[y]$?

I think the answer is yes, but didn't find a proof so far.

The answer is yes for chordal graphs due to simplicial vertices.

But for 4-chordal graphs, I did not see any result in this direction. There are generalizations of simplicial vertices of $k$-chordal graphs due to Krithika, Mathew, Narayanaswamy and Sadagopan (2013) and Chavatal, Rusu and Sritharan (2002), but these notions of simplicial vertices do not answer the question as far as I see ...