Termination of the modifying-module enlargement process

Determine whether the process of enlarging a modifying module by adjoining a modifying summand not already in its additive closure necessarily terminates, thereby establishing the existence of a maximal modifying module, or equivalently a maximal modification algebra, in the general setting considered.

Background

The paper defines a modifying module over a Cohen–Macaulay ring and calls it maximal modifying when every modifying direct-sum enlargement has no genuinely new summand. Starting from a non-maximal modifying module, one may repeatedly enlarge it by adjoining a modifying module outside its additive closure. The authors explicitly state that they do not know whether this process terminates, so the general existence of maximal modifying modules remains unresolved.