The Gotzmann threshold polynomiality conjecture for powers of $x_2$

Let $R_n=K[x_1,\dots,x_n]$ be the polynomial ring in $n$ variables over a field $K$, and let $\tau_n(x_2^d)$ denote the Gotzmann threshold of $x_2^d$ in $R_n$.

Gotzmann threshold polynomiality conjecture. For all $n \ge 3$, $\tau_n(x_2^d)$ is a polynomial of degree $2^{n-2}$ in $d$ with dominant term

$$\frac{d^{2^{n-2}}}{2^{2^{n-2}-1}}=2(d/2)^{2^{n-2}}.$$

Equivalently,

$$\lim_{d \to \infty} \frac{\tau_n(x_2^d)}{\binom{\tau_{n-1}(x_2^d)}2}=1.$$

The conjecture extends the explicit formulas known for $n=3,4,5$ and predicts the leading asymptotic behavior of the Gotzmann threshold in all higher dimensions. Determining $\tau_n(x_2^d)$ for general $n \ge 6$ remains open.