Conjectural asymptotics for numerical-semigroup sequences

Let $s_g$, $\hat n_g$, and $\mathsf{r}_{1.51}$ be the sequences and constant defined in the source, where $g$ tends to infinity. Conjectural asymptotics. The paper gives three successively stronger conjectures:

$$\lim_{g \to \infty} \hat n_g^{1/g}=\lim_{g\to\infty}s_g^{1/g}=\mathsf{r}_{1.51}.$$

$$\lim_{g \to \infty}\frac{\hat n_g}{\hat n_{g-1}}=\lim_{g\to\infty}\frac{s_g}{s_{g-1}}=\mathsf{r}_{1.51}.$$

3. There exist constants $\alpha$ and $\alpha'$ (possibly equal) such that

$$\hat n_g=g^{\alpha+o(1)}\mathsf{r}_{1.51}^g\qquad\text{and}\qquad s_g=g^{\alpha'+o(1)}\mathsf{r}_{1.51}^g.$$

These claims concern the conjectural growth of the auxiliary sequences $s_g$ and $\hat n_g$; the source motivates them by numerical extrapolation and states that each successive claim implies the next.