Shur's Restivo–Salemi conjecture for power-free languages

Let $\Sigma_k$ be a $k$-letter alphabet, and let $L_{k,\alpha}$ denote the language of $\alpha$-power-free finite words over $\Sigma_k$. For a language $L$, write ${\sf ext}(L)$ for its extendable factors. The language $L_{k,\alpha}$ has the Restivo–Salemi property when every pair of words $u,v\in {\sf ext}(L_{k,\alpha})$ can be joined by a word $w$ such that $uwv\in {\sf ext}(L_{k,\alpha})$. Shur's Restivo–Salemi conjecture. All power-free languages satisfy the Restivo–Salemi property. The property would imply that each power-free language has an infinite recurrent word whose subword complexity has the same growth rate as the language; the source attributes the conjecture to Shur and reports that it is based on extensive numerical studies, but gives no resolution.