Mirror invariance conjecture for the Thue–Morse two-block language

Let $x^{(00)}$ be the fixed point of the Thue–Morse two-block substitution, and let $L_{\rm\sc TM}$ be the language of all finite words occurring in $x^{(00)}$. A set of binary words is mirror invariant if, whenever it contains $w_1\dots w_m$, it also contains $(1-w_1)\dots(1-w_m)$. Mirror invariance conjecture. The language $L_{\rm\sc TM}$ is mirror invariant. This is contrasted with the fact that the language is not reversal invariant; proving mirror invariance is difficult because mirror images of occurring words may first appear only after many substitution iterations.