Compressed-graph invariance for similarly factored ideals in UFD quotients

Let $D_1$ and $D_2$ be unique factorization domains, and let $I_1\subset D_1$ and $I_2\subset D_2$ be nonmaximal, nontrivial ideals. Suppose

$$I_1=\bigcup_{\alpha\in\Lambda}\left\langle\prod_{k=1}^{Q_\alpha}p_{\alpha_k}^{s_{\alpha_k}}\right\rangle.$$

Assume that $I_2$ has an expression with the same index set, numbers of factors, and exponents,

$$I_2=\bigcup_{\alpha\in\Lambda}\left\langle\prod_{k=1}^{Q_\alpha}q_{\alpha_k}^{s_{\alpha_k}}\right\rangle.$$

Here the $p_{\alpha_k}$ and $q_{\alpha_k}$ are the corresponding irreducible factors.

Compressed-graph invariance conjecture. Under these hypotheses,

$$\Gamma_C(D_1/I_1)\cong\Gamma_C(D_2/I_2).$$

Thus the compressed zero-divisor graph is conjectured to depend only on the displayed factorization pattern of the ideal, not on the particular UFDs or irreducibles.