The LGL stretching property conjecture

Let $[a,b]$ be an interval, let $L=L_{[a,b]}:[a,(a+b)/2]\to[a,b]$ be given by $L(x)=2x-a$, and let $\{{\cal G}^{\mathrm{LGL}}_N\}_{N\in\mathbb{N}}$ be the family of Legendre–Gauss–Lobatto grids. A family of symmetric grids has property $\mathbf{Str}_{\bar N}$ if, for every $N\leq\bar N$, every interval $I\in\mathcal{P}({\cal G}_N)$ with $I\subset(a,a+(b-a)/4]$, and every $I'\in\mathcal{P}({\cal G}_N)$ such that $L(I)\cap I'\neq\varnothing$, one has $|I'|\leq|L(I)|$. The LGL stretching property conjecture. The family $\{{\cal G}^{\mathrm{LGL}}_N\}_{N\in\mathbb{N}}$ has property $\mathbf{Str}_{\bar N}$ for every $\bar N\in\mathbb{N}$. This is supported by numerical verification through order $\bar N=2000$; the assertion for all orders remains open.