  We prove a H\"older-type inequality for the Hausdorff distance between
Lagrangians with respect to the Lagrangian spectral distance or the
Hofer-Chekanov distance in the spirit of Joksimovi\'c-Seyfaddini
[arXiv:2207.11813]. This inequality is established via methods developped by
the first author [arXiv:2204.02468, arXiv:2108.00555] in order to understand
the symplectic geometry of certain collections of Lagrangians under metric
constraints.
