The conjectured bases for hereditary properties of tournaments up to 1.755

Let $cmathcal{X}$ be the set of real numbers $c$ for which some hereditary property of tournaments has exponential base $c$, meaning that $clim_{n\to\infty}|\mathcal{P}_n|^{1/n}=c$. Let $c'\simeq1.755$ be the largest real root of

$$x^4=x^3+x^2+1,$$

and let

$$A=\left\{x:x\text{ is the largest real root of }x^{k+1}=x^k+x^{k-2}+x^{k-3}+\cdots+1\text{ for some }2\leq k\in\mathbb{N}\right\}.$$

Conjecture on the possible bases.

$$\mathcal{X}\cap[0,c']=\{0,c'\}\cup A.$$

The claim describes the expected complete set of exponential bases in this interval; the paper gives constructions motivating the listed values, but the classification remains open.