The general structure conjecture for higher level Zhu algebras of the Heisenberg vertex operator algebra

Let $M_a(1)$ denote the Heisenberg vertex operator algebra with parameter $a$, and let $A_n(M_a(1))$ be its level-$n$ Zhu algebra. For $n>2$, let $p(n)$ be the number of unordered partitions of $n$ into nonnegative integers, and let $M_{p(n)}(\mathbb{C})$ denote the algebra of $p(n)\times p(n)$ matrices.

Higher level Zhu algebra conjecture. One has

$$A_n(M_a(1)) \cong A_{n-1}(M_a(1)) \oplus \left( \mathbb{C}[x] \otimes M_{p(n)}(\mathbb{C})\right).$$

The conjecture gives the expected recursive structure of the higher level Zhu algebras for the Heisenberg vertex operator algebra. The source states that it was proven for $n=1$ elsewhere and for $n=2$ in the paper, while the general case $n>2$ remains open.