In [Math. Proc. Cambridge Phil. Soc. 64 (1968), No. 2, 251-264], P.M. Cohn famously claimed (without proof) that a commutative domain is atomic if and only if it satisfies the ascending chain condition on principal ideals (ACCP). Some years later, A. Grams [Math. Proc. Cambridge Phil. Soc. 75 (1974), No. 3, 321-329] showed (by way of a counterexample) that Cohn's statement is wrong: Every commutative domain satisfying the ACCP is atomic, but not the other way around. It is therefore natural to wonder if Cohn's claim can be, in a sense, fixed by providing a sensible characterization (say, of a somewhat ideal-theoretic flavour) of when a commutative domain (or, more generally, a cancellative commutative monoid) is atomic.

Q. I was confirmed that the question is open and well known (at least in some circles), but haven't found any solid evidence in support of this. Do you happen to know of any papers, conference proceedings, etc. (the older, the better) where the question, although somewhat vague in its formulation, is explicitly stated?

Let me recall that a (multiplicatively written) monoid $H$ is atomic if every non-unit factors as a product of atoms (i.e., elements that cannot be written as a product of two non-units); and it satisfies the ACCP if there does not exist any (infinite) sequence $x_1, x_2, \ldots$ in $H$ such that $Hx_i H \subsetneq Hx_{i+1} H$ for all $i \in \mathbb N^+$. Accordingly, a domain is atomic if so is the multiplicative monoid $R^\bullet$ of its non-zero elements; and satisfies the ACCP if so does $R^\bullet$.

Edit 1. In the last lines of p. 3 in A. Geroldinger and F. Halter-Koch's monograph Non-Unique Factorizations. Algebraic, Combinatorial and Analytic Theory, the authors cite Grams' example and add that "Up to now, there is no satisfactory ideal-theoretic characterization of atomic domains." The book dates back to 2006, but the question is likely ten to thirty years older.

Edit 2 (addressing a comment by მამუკა ჯიბლაძე). Apart from Grams' counterexample to Cohn's statement, further examples of atomic commutative domains that do not satisfy the ACCP were given by A. Zaks [J. Algebra 80 (1982), 223-231] (where the author considers certain quotients of a polynomial ring in infinitely many variables and proves that they are atomic without the ACCP) and M. Roitman [J. Pure Appl. Algebra 87 (1993), 187-199] (where the author famously shows the existence of an atomic commutative domain $R$ such that the polynomial ring $R[X]$ is not atomic, incidentally producing an atomic commutative domain without the ACCP). More recently, further examples were provided by J. G. Boynton and J. Coykendall [J. Pure Appl. Algebra 223 (2019), 619-625] (where the authors use pullbacks of commutative rings to construct large families of atomic commutative domains without the ACCP) and F. Gotti and B. Li [https://arxiv.org/abs/2111.00170] (where, among other things, the authors construct what is probably the first-ever example of an atomic commutative monoid ring that does not satisfy the ACCP).

It is definitely much easier to construct cancellative commutative monoids without the ACCP. For instance, S.T. Chapman et al. prove in Corollary 4.4 of [Amer. Math. Monthly 128 (2021), No. 4, 302-321] that, if $r$ is a non-zero rational number smaller than $1$ whose numerator is not $1$, then the cyclic Puiseux monoid generated by $r$ (i.e., the submonoid of the additive group of the rational field $\mathbb Q$ generated by $r, r^2, \ldots$) is atomic but does not satisfy the ACCP.