Zero density in the easily bored sequence

For a finite binary word $w$ and $x\in\{0,1\}$, let $r_w(x)$ be the largest integer $r\ge 1$ for which $wx$ ends in $u^r$ for some nonempty word $u$, and let $\ell_w(x)$ be the largest length $|u|$ among witnesses for that maximal $r$. Define $b_1=0$; for $m\ge 2$, let $b_m$ be the bit $x$ that minimizes $(r_{b_1\cdots b_{m-1}}(x),\ell_{b_1\cdots b_{m-1}}(x))$ in lexicographic order. Does the limit $\lim_{N\to\infty}N^{-1}|\{1\le m\le N:b_m=0\}|$ exist and equal $1/2$?