Bos's cubic and quartic simplex interpolation nodes

Let $S_d \subset \mathbb{R}^d$ be a nondegenerate simplex, with points written in barycentric coordinates. In the lists below, each type of node stands for all distinct permutations of its coordinates, padded with zeros to $d + 1$ coordinates, and a type with more than $d + 1$ coordinates is omitted. The cubic set $F_{3,d}$ consists of the types $$(1), \qquad (t, 1 - t), \qquad (\tfrac13, \tfrac13, \tfrac13), \qquad t = \frac{1 + 1/\sqrt{5}}{2},$$ and the quartic set $F_{4,d}$ of the types $$(1), \qquad (\tfrac12, \tfrac12), \qquad (s, 1 - s), \qquad (a, b, b), \qquad (\tfrac14, \tfrac14, \tfrac14, \tfrac14),$$ where $s = \bigl(1 + \sqrt{3/7}\bigr)/2$, $a = (4 + \sqrt{5})/11$ and $b = (7 - \sqrt{5})/22$. For $k = 3, 4$, $F_{k,d}$ is unisolvent for the real polynomials of total degree at most $k$ on $\mathbb{R}^d$, a space of dimension $N_{k,d} = \binom{d+k}{k}$; let $\ell_x^{(k,d)}$, $x \in F_{k,d}$, be its Lagrange polynomials. A unisolvent set is a Fejér set if the sum of the squares of its Lagrange polynomials is at most $1$ on $S_d$, and a Fekete set if it maximizes the absolute value of the Vandermonde determinant (in any fixed basis) among all configurations of the same number of points in $S_d$. Bos showed that a Fejér set is a Fekete set (via the Kiefer–Wolfowitz equivalence theorem), and noted that the Lagrange polynomials of a Fekete set are bounded by $1$ in absolute value on $S_d$. He proved by computer algebra that $F_{3,d}$ is a Fejér set for $1 \le d \le 28$. $F_{4,2}$ is a Fejér set (Bos, 1983), while for $d = 4$ the sum of squares of the Lagrange polynomials of $F_{4,4}$ exceeds $1$ at the centroid of $S_4$; for $3 \le d \le 7$ all Lagrange polynomials of $F_{4,d}$ are bounded by $1$ in absolute value on $S_d$.

Problem. Is each of the following true? (1) For every $d \ge 1$ and $z \in S_d$, $\sum_{x \in F_{3,d}} |\ell_x^{(3,d)}(z)|^2 \le 1$. (2) For every $z \in S_3$, $\sum_{x \in F_{4,3}} |\ell_x^{(4,3)}(z)|^2 \le 1$. (3) For every $d \ge 1$, $F_{4,d}$ maximizes the absolute value of the Vandermonde determinant among all configurations of $N_{4,d}$ points in $S_d$. (4) For every $d \ge 1$ and $z \in S_d$, $\sum_{x \in F_{4,d}} |\ell_x^{(4,d)}(z)|^4 \le 1$.