Monotonicity and threshold conjecture for optimal spreading under fixed-size access

In the fixed-size access model, let $N$ be the number of nodes, $m$ the redundancy level, $r$ the number of accessed nodes, and $b1$ the spreading parameter, with minimal spreading corresponding to $b1=1$ and maximal spreading corresponding to the largest feasible value of $b1$. Fixed-size spreading conjecture. Given $N$ and $m$, the optimal $b1$ increases as $r$ increases. Moreover, for every $N$ and $m$, there exists a $b3\frac{b6}{(1,N)}$ such that minimal spreading is optimal for all $rb7b3$, and there exists a $b6b5b6b5(1,N)$ such that maximal spreading is optimal for all $rb7b6$. These assertions would characterize low-access and high-access regimes for the service-rate-maximizing allocation, complementing the paper's sufficient conditions for minimal-spreading optimality and non-optimality.