Noether inequality for irregular threefolds of general type

Let X be a smooth irregular $3$-fold of general type over C. We prove that the optimal Noether inequality vol(X) ≥ 4/3pg(X) holds if pg(X) ≥ 16 or if X has a Gorenstein minimal model. Moreover, when X attains the equality and pg(X) ≥ 16, its canonical model can be explicitly described.