Is 2 the optimal worst-case approximation ratio of LPT on USP-C?

Ascertain whether the worst-case approximation ratio of the Longest Processing Time First heuristic on the Uniform Machine Scheduling Problem with Processing Set Restrictions (USP-C) is 2; either prove that 2 is tight or provide a strictly smaller upper bound.

Background

The paper positions DWP as a special case of USP-C, where each job can only be processed by a subset of machines. The authors construct a simple instance in USP-C for which LPT attains an approximation ratio approaching 2, contrasting with an optimal schedule that completes more quickly.

They explicitly note that whether this 2-approximation example is tight for LPT on USP-C remains unknown, highlighting a fundamental gap in understanding LPT's worst-case performance under processing set restrictions.