Taillard's asymptotic tightness conjecture for permutation flowshop lower bounds

Let $M$ be a fixed number of machines and let $LB=\max\{LB_M^+,LB_J\}$, where $LB_M^+$ and $LB_J$ are the machine-load and job-length lower bounds for the permutation flowshop scheduling problem, respectively. Let $OPT$ denote the optimal makespan, and interpret the probability over the processing-time instance distribution used for the problem.

Taillard's conjecture. For a fixed number of machines $M$,

$$\lim_{N\to\infty}\mathbb{P}(LB=OPT)=1.$$

The conjecture asserts that, as the number of jobs tends to infinity while the number of machines remains fixed, the lower bound $LB$ is almost surely tight. Its resolution depends on the precise random-instance model for processing times, which is not specified in the supplied statement.