Complexity of deciding constant-bounded mim-width

Determine the computational complexity of deciding whether the mim-width or linear mim-width of a graph is at most 1, or at most any other one-digit constant.

Background

The paper proves that Mim-Width, Sim-Width, One-Sided Mim-Width, and their linear variants are NP-complete even when the width is bounded by a universal constant, using a reduction with an upper bound of 1211. The authors note that their bounds do not resolve the substantially sharper threshold of one, or even any one-digit constant. Thus, the complexity of deciding whether a graph has mim-width or linear mim-width at most such a small fixed constant remains unresolved.