The monotonicity conjecture for LGL interval quotients

Let $\xi^N_0<\xi^N_1<\cdots<\xi^N_N$ be the LGL points, let $\Delta^N_k=[\xi^N_k,\xi^N_{k+1}]$, and define $q_k^N=|\Delta^N_k|/|\Delta^N_{k-1}|$ for the relevant indices. The family has property $\mathbf{MQ}_{\bar N}$ if $q_k^N\leq q_k^{N+1}$ for $N<\bar N$ and fixed $k$, and $q_k^N\geq q_{k+1}^N$ for $1\leq k\leq\lfloor(N-3)/2\rfloor$ and $N\leq\bar N$. The LGL quotient monotonicity conjecture. The LGL grids have property $\mathbf{MQ}_{\bar N}$ for every $\bar N\in\mathbb{N}$. Numerical evidence verifies the property through $\bar N=2000$; its validity for all orders remains open.