Probabilistic computation of Jacobson form from a diagonal form

Let $R$ be a simple Euclidean domain and let $M$ be a square matrix in diagonal form

$$M=\operatorname{Diag}(m_1,\ldots,m_r).$$

The quantity $\sum\deg(m_i)$ is an invariant of the module $R^{r\times r}/M$. Choose polynomials $p_i$ of degree at most $\deg(m_i)-1$ with random coefficients in $A$, and let $c\in R$ generate the left annihilator ideal of the vector $[p_1,\ldots,p_r]^T$ in $R^{r\times r}/M$. Jacobson-form computation conjecture. If $\deg(c)=\sum\deg(m_i)$, then $\operatorname{Diag}(1,\dots,1,c)$ is a Jacobson form of $M$; otherwise, repeating the procedure with another random set of polynomials eventually computes the Jacobson form. This gives the invariant degree sum as a certificate for a probabilistic approach. The conjecture proposes a practical method for computing Jacobson form from a diagonal form over simple Euclidean domains; the supplied text gives no resolution or evidence establishing termination of the repeated random procedure.