Lex or RevLex minimality conjecture for degree-two subalgebras

Let $\mathbb{k}[x_1,\ldots,x_n]$ be a polynomial ring. For a positive integer $u$ with $\binom{n}{2}<u\leq\binom{n+1}{2}$, let $\operatorname{Lex}(u)$ and $\operatorname{RevLex}(u)$ be the sets of the $u$ greatest degree-two monomials in the Lex and RevLex orders. Lex or RevLex minimality conjecture. One of the algebras

$$\mathbb{k}[\operatorname{Lex}(u)]\quad\text{or}\quad\mathbb{k}[\operatorname{RevLex}(u)]$$

has the minimal Hilbert function among subalgebras of $\mathbb{k}[x_1,\ldots,x_n]$ generated by $u$ forms of degree two. In the supplied text, this assertion is stated as proved by a computation of multiplicities in Mathematica, so its database status is solved rather than open.