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on Zero density in the easily bored sequence

The statement

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 $1/2$ ?

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.