The degree-four Lagrange-$\ell^4$ bound conjecture for real simplices

Let $S_d$ be the real $d$-simplex, let $F_4$ be the set of $N=\binom{d+4}{4}$ points constructed in the paper, and let $\ell_j$ be the associated fundamental Lagrange polynomials. The degree-four Lagrange-$\ell^4$ bound conjecture. For all dimensions $d\geq 1$ and all ${\bf x}\in S_d$, one has

$$\sum_{j=1}^N\ell_j^4({\bf x})\leq 1,$$

or, equivalently,

$$\|{\bm \ell}({\bf x})\|_4\leq 1.$$

Numerical computations support this for dimensions $d=3,4,5$, but the assertion for all dimensions is open.