If ( A X X∗ B ) ∈M2(Mn) is positive semidefinite, Lin [7] conjectured that 2s j(Ψ(X)) ≤ s j(Ψ(A) + Ψ(B)), j = 1, . . . ,n, and s j(Ψ(X)) ≤ s j(Ψ(A)]Ψ(B)), j = 1, . . . ,n, where the linear map Ψ : X 7→ 2tr(X)In − X and s j(·) means the j-th largest singular value. In this paper, we reprove that ( Ψ(A) Ψ(X) Ψ(X∗) Ψ(B) ) is PPT by using an alternative approach and prove the above singular value inequalities hold for the linear map Ψ1 : X 7→ (2n + 1)tr(X)In − X