The run-structure conjecture for words of minimal subword entropy

For $k \geq 2$, let $w$ be a binary word of length $n \geq 1$ achieving the minimal subword entropy $\min S_{\mathrm{sw}}^{(k)}(n)$. A run is a maximal consecutive block of equal letters. Run-structure conjecture. Except for finitely many values of $n$, the longest run in $w$ has length $3$, and the average run length converges as $n \to +\infty$.

The claim is motivated by computations showing that minimizers mostly contain runs of lengths $1$, $2$, and $3$, while long runs appear to increase subword entropy. Its status is open.