Compressed zero-divisor graph description for ideals in UFDs

Let $D$ be a unique factorization domain and let $I$ be an ideal with a minimal expression

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

Let $\"]\phi:D\to D/I$ be the canonical map, and let $\"]\mathfrak B$ be the zero-divisor basis of $D/I$, consisting of suitable nonassociate divisors of the generators. The compressed zero-divisor graph has vertices represented by annihilator-equivalence classes of elements $\"]\phi(\prod_i d_i)$ with each $d_i\in\mathfrak B$.

Compressed-graph structure conjecture. An edge joins the vertices represented by $\"]\phi(\prod_i d_i)$ and $\"]\phi(\prod_i b_i)$ if and only if

$$n_\alpha\mid\prod_i d_i b_i$$

for some $\"]\alpha\in\Lambda$.