Lower bound for the size of simplex t-designs

Let $N_*(d,t)$ be the minimal number of elements in a $t$-design in the $d$-point simplex, and let $\binom{a}{b}$ denote a binomial coefficient. Simplex-design lower-bound conjecture. The minimal number of elements satisfies

$$N_*(d,t) \geq \binom{d+\frac{t}{2}-1}{\frac{t}{2}}.$$

This proposed bound generalizes the rank-based lower bound established in the paper for $t=4$ and is intended to describe the growth of the minimum design size beyond the paper's main focus; it remains open in general.