The degree-four Fekete-set conjecture for real simplices

Let $S_d$ be the real $d$-simplex, let $F_4$ be the set of $N_4=\binom{d+4}{4}$ points constructed in the paper, and let $\ell_i$ be its fundamental Lagrange polynomials. The degree-four Fekete-set conjecture. The sets $F_4$ are Fekete sets for every dimension $d=1,2,\ldots$. This is known in dimension $d=2$, while the preceding inequality is conjectured in dimension $d=3$ and fails in dimension $d=4$; the general Fekete-set assertion remains open.