  We show the existence of cones over 8-dimensional rational spheres at the
boundary of the Mori cone of the blow-up of the plane at $s\geq 13$ very
general points. This gives evidence for De Fernex's strong $\Delta$-conjecture,
which is known to imply Nagata's conjecture. This also implies the existence of
a multitude of good and wonderful rays as defined in
Ciliberto-Harbourne-Miranda-Ro\'e 2013.
