Cossu–Zanardo necessity conjecture for idempotent factorizations

Let $D$ be the quadratic-ring parameter, let $p$ be as in the Cossu–Zanardo setting, let $I_p(D)$ be the set used in the source, and let $z\in I_p(D)$. Consider a matrix

$$\begin{pmatrix}p&z\\ \bar z&k\end{pmatrix}.$$

Cossu–Zanardo necessity conjecture. If this matrix can be written as a product of two idempotent matrices, then it must satisfy the Cossu–Zanardo conjecture.

The conjecture is motivated by examples of matrices that fail the Cossu–Zanardo factorization and consequently cannot be written as products of two idempotent matrices. The supplied context does not define $I_p(D)$ or state a resolution.