Quadratic Gröbner bases for symbolic Rees algebras of squarefree Veronese ideals

Determine for which positive integers n and d the defining ideal of the symbolic Rees algebra of the squarefree Veronese ideal I_{n,d} has a quadratic Gröbner basis with respect to some monomial order.

Background

For a squarefree Veronese ideal I_{n,d}, the paper proves that the initial ideal of the defining ideal of its symbolic Rees algebra is generated in degrees at most three for a specified monomial order. The authors also show that quadratic generation does not hold in general for that order, giving I_{4,3} as an example whose initial ideal has a cubic generator.

Question 3.8 asks when a different monomial order might yield a quadratic Gröbner basis. This is a concrete unresolved problem concerning the algebraic presentation of symbolic Rees algebras of squarefree Veronese ideals.