Let $\Lambda_2 = \mathbb{Z}_p[[T_1,T_2]]$ be the power series ring over $\mathbb{Z}_p$ in two variables, i.e., $\Lambda_2$ is a regular local ring of dimension 3. Let $M$ be the quotient of an elementary torsion $\Lambda_2$-module of the form $E = \bigoplus_i \Lambda_2/(f_i)$. Suppose that $M$ is a finitely generated $\Lambda_2$-module which is annihilated by two relatively prime elements (i.e., $M$ is pseudo-null). This means that one can choose elements $h_1, h_2$ in the annihilator ideal $\text{Ann}(M)$ of $M$ such that the Krull dimension of $\Lambda_2/(h_1,h_2)$ is 1.

Question: If the Krull dimension of $\Lambda_2/\text{Ann}(M)$ happens to be also 1, can one choose $h_1$ and $h_2$ contained in $\text{Ann}(M)$ such that some power of the annihilator ideal is contained in the ideal $(h_1, h_2)$?