Conjecture on optimal allocation under fixed-size access

Let $r$ denote the number of accessed nodes, let $\alpha$ be the allocation parameter, and let $\mu_s(\alpha)$ denote the average service rate under fixed-size access. Fixed-size access conjecture. When $r$ is small, $\mu_s(1)$ is more likely to be the maximum; when $r$ is large, the maximum of $\mu_s(\alpha)$ is not attained at $1$, and the optimal $\alpha$ is increasing with $r$. The conjecture describes the qualitative dependence of the optimal allocation on the number of accessed nodes; the preceding analysis establishes behavior in two parameter regions but leaves an intermediate gap undecided.