The even-base min-entropy formula conjecture for entropoid power shapes

Let $\mathbb{E}_{p^2}$ be an entropoid and let $g\in\mathbb{E}^*_{(p-1)^2}$ generate its maximal multiplicative subgroupoid. For every even base $\mathfrak{b}=2b_1<b_{\max}$, let $n(\mathfrak{b},i)=\max n_{ij}$. The even-base min-entropy conjecture.

$$n(\mathfrak{b},i)=(\mathfrak{b}-1)\left((\mathfrak{b}-1)^{i-1}-(\mathfrak{b}-2)^{i-1}\right),$$

and consequently

$$H_{\infty}(\xi_i)=1-\left(\frac{\mathfrak{b}-2}{\mathfrak{b}-1}\right)^{i-1}.$$

The formula formalizes the observed even-versus-odd base pattern in the distribution of bracketing shapes; the paper reports it as an observed conjectural relation rather than proving it.