Conjecture on the non-monomial associated prime of $I_n(B)$

Let $I_n(B)$ be the ideal associated with the matrix $B$ in the construction of the Rees algebra, and let $\mathcal{P}$ denote the non-linear part of the defining ideal of that Rees algebra. When the $G_n$ condition is not satisfied, $I_n(B)$ is not prime. Non-monomial associated-prime conjecture. The ideal $\mathcal{P}$ is the only associated prime of $I_n(B)$ that is not generated by monomials. The preceding discussion indicates that $\mathcal{P}$ is expected to be a minimal prime of $I_n(B)$; the claim concerns the uniqueness of the associated prime with this non-monomial property.