Central-material conjecture for optimal one-dimensional refractive indices

Let $n_\star(x)$ be a locally optimal one-dimensional refractive index on the interval $[0,L]$, taking the values $n_+$ and $n_-=1$ on its material intervals. The point $x=L/2$ is the center of the interval. Central-material conjecture. The optimal refractive index satisfies

$$n_\star(L/2)=n_+.$$

The conjecture is motivated by the expectation that the mode energy concentrates where the refractive index is large and, in particular, near the center of the domain. No resolution is given in the source.