Cossu–Zanardo conjecture on factorizations of quadratic-ring matrices

Let $D$ be a square-free integer, let $p$ be a prime integer that is irreducible but not prime in $\mathbb{Z}[\sqrt{D}]$, and let $z\in\mathbb{Z}[\sqrt{D}]$ be such that $\langle p,z\rangle$ is a non-principal ideal. Define

$$A(p,z)=\begin{pmatrix} p&z\\ \bar{z}&\sfrac{\lVert z\rVert}{p} \end{pmatrix}.$$

Cossu–Zanardo conjecture. There exist $a,b,c\in\mathbb{Z}[\sqrt{D}]$ with $a(1-a)=bc$ such that

$$A(p,z)=\begin{pmatrix} a&b\\ c&1-a \end{pmatrix}\begin{pmatrix} \bar{a}&\bar{c}\\ \bar{b}&1-\bar{a} \end{pmatrix}.$$

Cossu and Zanardo introduced this conjecture for singular matrices over quadratic integer rings. The paper states that some classes of matrices affirm it while others oppose it, and that the complete classification of singular $2\times2$ matrices over these rings remains open.