Idempotent factorization on some matrices over quadratic integer rings

In 2020, Cossu and Zanardo raised a conjecture on the idempotent factorization on singular matrices in the form $\begin{pmatrix} p&z\\ \bar{z}&\sfrac{\lVert z\rVert}{p} \end{pmatrix},$ where $p$ is a prime integer which is irreducible but not prime element in the ring of integers $\mathbb{Z}[\sqrt{D}]$ and $z\in\mathbb{Z}[\sqrt{D}]$ such that $\langle p,z\rangle$ is a non-principal ideal. In this paper, we provide some classes of matrices that affirm the conjecture and some classes of matrices that oppose the conjecture. We further show that there are matrices in the above form that can not be written as a product of two idempotent matrices.