  We develop tools to prove D\'iaz and Park's sharpness conjecture (see [8])
for fusion systems admitting tame families of fusion subsystems (see Theorem
A). We use such tools to prove the conjecture for all Benson-Solomon fusion
systems (see Theorem B) thus completing the work started by Henke, Libman and
Lynd in [14, Theorems 1.1 and 1.4].
