Optimality of full replication for log-convex service distributions

Let $n$ be the number of servers, let $\bar{F}_X$ be the survival function of a task's service time, and let $\rX$ denote the service-time distribution. For a fork-join system, let $T$ be the job latency and $C$ be the total computational cost, with each job forked to $r$ servers. Full-replication optimality conjecture. For log-convex $F_X$ and any arrival rate $\lambda$, forking a job to all $n$ servers is optimal in both $\mathbb{E}[T]$ and $\mathbb{E}[C]$. The preceding discussion establishes the corresponding high-traffic behavior and observations suggest that the claim holds for every arrival rate, but no proof or resolution is supplied here.