Conical-vertex conjecture for extremal edge-minimizing systems

For integers $r\geq 3$ and $t\geq r$, let $e_{r,t}(s)$ and $e_{r,t}'(s)$ be the minimum number of edges in the underlying graph of an $(r,t)$-system and a maximal $(r,t)$-system, respectively, with family size $s$. An extremal system is one attaining the relevant minimum, and a vertex is conical if it is adjacent to every other vertex of its underlying graph. Edge-system conical-vertex conjecture. For all integers $r\geq 4$ and $t\geq r$, every underlying graph of an extremal system for $e_{r,t}(s)$ or $e_{r,t}'(s)$ has a conical vertex for large enough $s$. This predicts a conical structure for both edge-minimization problems, with and without maximality.