Monotonicity of worst-case bottleneck values for the $(\delta,k)$-PKRY algorithm

For each $m$, let $S_m$ be a star with $m$ children rooted at its centre vertex, chosen to maximise the length of the longest edge in the tree produced by the $(\delta,k)$-PKRY algorithm, with every child at unit distance from the centre and no pair of children at distance less than $1$. Define $S_{m+1}$ similarly with $m+1$ children. Let $b$ and $b'$ be the bottleneck values of the outputs of $(\delta,k)$-PKRY on $S_m$ and $S_{m+1}$, respectively. PKRY monotonicity conjecture. Then $b \geq b'$. This is presented as a monotonicity property for worst-case arrangements, and the source describes it as a stronger version of an earlier conjecture; no proof or resolution is supplied here.