Irreducibility of some nested Quot schemes

Let C be a smooth projective curve over C of genus g 1. Let E be a vector bundle on C of rank r and degree e. Given integers k₁,k₂,d₁,d₂ such that r>k₁>k₂>0, let Qk₁,k₂d₁,d₂(E) denote the nested Quot scheme which parametrizes pair of quotients [E F₁ F₂] such that Fᵢ has rank kᵢ and degree dᵢ. We show that these nested Quot schemes are integral, local complete intersection schemes when d₁ d₂ 0 or d₂ d₁ 0.