Sufficient stability condition for redundancy-$d$ systems with processor sharing

Let a redundancy-$d$ system use processor sharing (PS), and let job sizes have general distributions. Write $\tilde{\rho}$ for the corresponding system load parameter. Sufficient stability condition. The condition

$$\tilde{\rho} < 1$$

is sufficient for stability. The conjecture would extend fluid-limit stability results from head-of-the-line service disciplines to processor sharing and other non-head-of-the-line disciplines. The corresponding extension to general job-size distributions is described as beyond the scope of the paper and as a challenging open problem.