Let $A$ be the algebra of $n \times n$ upper triangular matrices over a field $K$ and $J$ its Jacobson radical. Call an ideal $I$ of $A$ admissible in case $I \subseteq J^2$. Let $X$ be the set of algebras of the form $A/I$ with $I$ admissible and call those algebras $n$-LNakayama algebras.

There are $C_{n-1}$ (Catalan number) $n$-LNakayama algebras.

Let $B$ be such an $n$-LNakayama algebra. Call this algebra syzygy-closed in case the injective dimension of $P_i$ is bounded by $i+1$ when $(P_i)_{i \geq 0}$ is the minimal projective resolution of $D(B)=Hom_K(B,K)$. (By a result of Auslander and Reiten this is equivalent to the subcategories $\Omega^{-i}(mod-A)$ all being extension-closed)

Question: How many $n$-LNakayama algebras are there that are syzygy closed?
For $n=1,...,9$ my computer said that the sequence starts as follows:
https://oeis.org/A264228 . I am highly surprised that such a complicated question might have an answer that exists in the OEIS.