GCD representatives for compressed zero-divisor graph classes in quotients of UFDs

Let $D$ be a unique factorization domain and let $I$ be an ideal expressed as a minimal union of principal ideals

$$I=\bigcup_{\alpha\in\Lambda}\langle n_\alpha\rangle.$$

Let $\"]\phi:D\to D/I$ be the canonical homomorphism, and let $\"][x]$ denote the annihilator-equivalence class of a nonzero zero-divisor $x$ in the compressed zero-divisor graph.

GCD-representative conjecture. For every $a\in D$,

$$[\phi(a)]=\left[\phi\left(\gcd\{a,n_\alpha\mid\alpha\in\Lambda\}\right)\right].$$

This would provide canonical gcd-type representatives for compressed zero-divisor graph classes in quotients by ideals with minimal principal-union expressions.