A weakly-2-generic which bounds a minimal degree

Jockusch showed that 2-generic degrees are downward dense below a 2-generic degree, but in the case of 1-generic degrees Kumabe, and independently Chong and Downey constructed a minimal degree computable from a 1-generic degree. We explore the tightness of these results. We solve a question of Barmpalias and Lewis-Pye by constructing a minimal degree computable from a weakly 2-generic one. The proof is rather novel since it is a computable full approximation construction where both the generic and the minimal degrees are ∆3 −∆2.