The strict lower-bound conjecture for binary subword entropy

Let $L_2$ denote the asymptotic binary subword-entropy constant, and let $\log_2$ be the logarithm to base $2$. Strict lower-bound conjecture.

$$L_2 > \log_2(3/2).$$

This conjecture is supported by exhaustive computations suggesting that the universal lower bound is not tight, although the available data are limited to small word lengths and the claim remains open.