The hereditary atomicity conjecture for integral domains

Let $R$ be an integral domain. It is hereditarily atomic if every subring of $R$ is atomic, and $R$ satisfies the ascending chain condition on principal ideals (ACCP) if every ascending chain of principal ideals in $R$ eventually stabilizes.

Hereditary atomicity conjecture. Every hereditarily atomic domain satisfies the ACCP.

This was conjectured by Hasenauer and the authors and concerns the relationship between atomicity of all subrings and stabilization of principal ideals. The conjecture is refuted: $cmathbb{Q}[x]$ satisfies the ACCP but contains the non-atomic subring $cmathbb{Z}+xcmathbb{Q}[x]$.