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Let (M , ω) be a compact connected Kähler manifold of complex dimension four and let [ χ ] ∈ H 1 , 1 (M ; R). We confirm the conjecture by Collins–Jacob–Yau [8] of the solvability of the deformed Hermitian–Yang–Mills equation, which is given by the following nonlinear elliptic equation ∑ i arctan ⁡ (λ i) = θ ˆ , where λ i are the eigenvalues of χ with respect to ω and θ ˆ is a topological constant. This conjecture was stated in [8] , wherein they proved that the existence of a supercritical C -subsolution or the existence of a C -subsolution when θ ˆ ∈ [ ((n − 2) + 2 / n) π / 2 , n π / 2) will give the solvability of the deformed Hermitian–Yang–Mills equation. Collins–Jacob–Yau conjectured that their existence theorem can be improved to θ ˆ > (n − 2) π / 2 , where n is the complex dimension of the manifold. In this paper, we confirm their conjecture that when the complex dimension equals four and θ ˆ is close to the supercritical phase π from the right, then the existence of a C -subsolution implies the solvability of the deformed Hermitian–Yang–Mills equation. [ABSTRACT FROM AUTHOR]