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Carlsen [‘
$\ast $-isomorphism of Leavitt path algebras over 
$\Bbb Z$’, Adv. Math. 324 (2018), 326–335] showed that any 
$\ast $-homomorphism between Leavitt path algebras over 
$\mathbb Z$ is automatically diagonal preserving and hence induces an isomorphism of boundary path groupoids. His result works over conjugation-closed subrings of 
$\mathbb C$ enjoying certain properties. In this paper, we characterise the rings considered by Carlsen as precisely those rings for which every 
$\ast $-homomorphism of algebras of Hausdorff ample groupoids is automatically diagonal preserving. Moreover, the more general groupoid result has a simpler proof.
