Nonexistence of d-point simplex 3-designs

Let $\Delta_d$ denote the $d$-point probability simplex, and let a simplex $t$-design be a collection of points in $\Delta_d$ whose moments agree with those of the uniform simplex measure through degree $t$. Nonexistence conjecture. There are no $d$-point simplex 3-designs in $\Delta_d$ for $d\geq 3$. This would extend the established nonexistence result for simplex 4-designs and is motivated by the known obstruction in dimension $d=3$ together with numerical evidence; the general case remains open.