In 2009, Shur published the following conjecture: Let $L$ be a power-free language and let $e(L)\subseteq L$ be the set of words of $L$ that can be extended to a bi-infinite word respecting the given power-freeness. If $u, v \in e(L)$ then $uwv \in e(L)$ for some word $w$. Let $L_{k,\alpha}$ denote an $\alpha$-power free language over an alphabet with $k$ letters, where $\alpha$ is a positive rational number and $k$ is positive integer. We prove the conjecture for the languages $L_{k,\alpha}$, where $\alpha\geq 5$ and $k\geq 3$.

翻译：2009年，Shur提出了如下猜想：设$L$为幂自由语言，$e(L)\subseteq L$表示$L中可扩展为满足给定幂自由条件的双无限词的词集。若$u, v \in e(L)$，则存在某个词$w$使得$uwv \in e(L)$。设$L_{k,\alpha}$表示包含$k$个字母的字母表上的$\alpha$-幂自由语言，其中$\alpha$为正有理数，$k$为正整数。我们证明了当$\alpha\geq 5$且$k\geq 3$时，该猜想对语言$L_{k,\alpha}$成立。