The exponential growth conjecture for generalized numerical semigroups

Let $d\in\mathbb{N}_{>0}$, let $F\in\mathbb{N}^d$, and let $N(F)$ denote the number of generalized numerical semigroups with Frobenius element $F$. For $d\ne 5$, define $d_1=\left\lceil\frac{d+1}{3}\right\rceil$ and

$$a_d=\left(\left(\sqrt{3}\right)^{\frac{1}{2^{d}}\sum_{i=d_1}^{d-d_1}\binom{d}{i}}\times \Big(2\Big)^{\frac{1}{2^d}\sum_{i=d-d_1+1}^{2d_1-1}\binom{d}{i}}\right),$$

while $a_5=\sqrt{2}$. The exponential growth conjecture. For each $d\in\mathbb{N}_{>0}$, $N(F)$ is of the magnitude of $a_d^{\|F\|}$. The paper establishes the bounds $a_d^{\|F-1\|}\leq N(F)\leq O\left(\|F\|b_d^{\|F\|}\right)$, with $\sqrt{2}\leq a_d\leq b_d<\sqrt{3}$ and both constants tending to $\sqrt{3}$ as $d\to\infty$. Determining whether the lower-bound base $a_d$ gives the actual exponential magnitude of $N(F)$ remains open.