The bounded indecomposable-syzygy conjecture for local Golod rings

Let $(R,\mathfrak{m},k)$ be a local Golod ring of embedding codimension $e$. An indecomposable-syzygy conjecture asserts that there is a set of at most $e+1$ indecomposable $R$-modules from which every syzygy of $k$ may be built as a direct sum.

This would generalize the phenomenon established in the paper for embedding codimension $2$, where at most three non-isomorphic indecomposable modules occur among the direct-sum decompositions of the syzygy modules of $k$. The statement is presented as an observation from experiments for embedding codimension greater than $2$; no proof or resolution is supplied here.