On the Gotzmann threshold of monomials

Let $R_n=K[x_1,\dots ,x_n]$ be the $n$-variable polynomial ring over a field $K$. Let $S_n$ denote the set of monomials in $R_n$. A monomial $u\in S_n$ is a $\mathit{\text{Gotzmann}}\mathit{\text{monomial}}$ if the Borel-stable monomial ideal $⟨u⟩$ it generates in $R_n$ is a Gotzmann ideal. A longstanding open problem is to determine all Gotzmann monomials in $R_n$. Given $u_0\in S_{n-1}$, its $\mathit{\text{Gotzmann}}\mathit{\text{threshold}}$ is the unique nonnegative integer $t_0={\tau }_n(u_0)$ such that $u_0{x}_n^t$ is a Gotzmann monomial in $R_n$ if and only if $t\ge t_0$. Currently, the function ${\tau }_n$ is exactly known for $n\le 4$ only. We present here an efficient procedure to determine ${\tau }_n(u_0)$ for all $n$ and all $u_0\in S_{n-1}$. As an application, in the critical case $u_0=x_2^d$, we determine ${\tau }_5(x_2^d)$ for all $d$ and we conjecture that for $n\ge 6$, ${\tau }_n(x_2^d)$ is a polynomial in $d$ of degree $2^{n-2}$ and dominant term equal to that of the $(n-2)$-iterated binomial coefficient $$\left(\genfrac{}{}{0px}{}{\left(\genfrac{}{}{0px}{}{\left(\genfrac{}{}{0px}{}{d}{2}\right)}{2}\right)}{}\right)\mathrm{.}$$