Bonanzinga–Eliahou conjecture on Gotzmann thresholds of powers of $x_2$

Let $R_n$ be the polynomial ring in $n$ variables, and let $\tau_n(x_2^d)$ denote the Gotzmann threshold of the monomial $x_2^d$. For $n\geq 3$, define the $(n-2)$-iterated binomial coefficient by

$$\begin{pmatrix} \binom{\binom{d}{2}}{2}\\ \cdots\\ 2 \end{pmatrix}.$$

Bonanzinga–Eliahou's conjecture. For $n\geq 3$, the Gotzmann threshold $\tau_n(x_2^d)$ is a polynomial of degree $2^{n-2}$ in $d$ with the same dominant term as the $(n-2)$-iterated binomial coefficient above, namely

$$2(d/2)^{2^{n-2}}.$$

The conjecture predicts the leading growth of the Gotzmann thresholds in arbitrary numbers of variables from the formulas known for $n\in\{3,4,5\}$. Determining exact formulas for general monomials is described as difficult, while this asymptotic polynomial behavior remains the proposed problem.