Exponential lower-bound conjecture for peeling sequences

Let $g_d(n)$ denote the minimal number of peeling sequences of $n$ points in $\mathbb{R}^d$. Here $c$ and $\epsilon$ are positive constants.

Peeling-sequence growth conjecture. The minimal number of peeling sequences of $n$ points in $\mathbb{R}^d$ is at least

$$c(d+1+\epsilon)^n,$$

so that

$$g_d(n)=\Omega((d+1+\epsilon)^n).$$

The conjecture proposes an improvement over the trivial lower bound for the minimal number of peeling sequences. Its motivation comes from the Fractional Erdős–Szekeres Theorem, while the source states that the lower bound remains open.