The polynomial-exponent enumeration conjecture for 2-level polytopes

Let $\ell(d)$ be the number of combinatorially distinct $2$-level polytopes in dimension $d$. Enumeration conjecture for 2-level polytopes. The number $\ell(d)$ of combinatorially distinct $2$-level $d$-polytopes satisfies

$$\ell(d)\leq 2^{\operatorname{poly}(d)}.$$

Known constructions give the lower bound $\ell(d)\geq 2^{\Omega(d^2)}$, including constructions from stable set polytopes of bipartite graphs. The proposed polynomial-exponent upper bound remains open.