Simplex design factorial-size conjecture

Let $\Delta^{d-1}=\{\vb{x}\in\mathbb{R}^d:\sum_{i=1}^d x_i=1,\ x_i\geq 0\}$ be the probability simplex, and let $\sigma$ be the normalized measure supported on it and constant there. Consider the affine space

$$\mathbb{R}^d/\left(\sum_{i=1}^d x_i=-1\right).$$

A finite subset $X$ of this space is a $t$-design for $\sigma$ when its discrete averages reproduce the corresponding $\sigma$-averages for polynomials of degree at most $t$.

Simplex design factorial-size conjecture. There exists a $t$-design $X\subset \mathbb{R}^d/\left(\sum_{i=1}^d x_i=-1\right)$ for $\sigma$, with $t\leq d$, such that

$$\lvert X\rvert=\frac{d!}{(d-t+1)!}.$$

The conjecture proposes a factorial-size construction extending the previously established simplex $3$-design and pseudodesign constructions. Its validity beyond the dimensions and strengths covered by the preceding theorem remains open.