Monotonicity of worst-case bottleneck values for degree-bounded spanning trees

Let $v$ be a root vertex with children $\{v_1, \dots, v_k\}$, where $k \geq 2$, the angles between children with respect to $v$ are at least $60^{\circ}$, and all children have radial distance $1$. Let $P$ be the optimal bottleneck path that starts at $v$ and visits all children, and suppose that the children are placed to maximise the bottleneck length $b$ of $P$. Likewise, let $w$ be a root vertex with children $\{w_1, \dots, w_{k+1}\}$, with pairwise angles at least $60^{\circ}$ and radial distances equal to $1$. Let $P'$ be the optimal bottleneck path that starts at $w$ and visits all children, and suppose that the children are placed to maximise the bottleneck length $b'$ of $P'$. Monotonicity conjecture. Then $b \geq b'$. This asserts that adding a child cannot increase the worst-case bottleneck value. The claim is motivated by analytical and experimental point sets, but no proof or resolution is supplied here.