Locating parameters where the principal asymptotic class equals the Gamma class

Determine whether there exists a parameter q in the Kähler moduli space M_X such that c1(X)⋆_q has a simple rightmost eigenvalue and [A_X(τ)]=[Γ_X]; ascertain whether such τ can be taken in H^2(X,R); and characterize the region in H^2(X) where [A_X(τ)]=[Γ_X] holds.

Background

The principal asymptotic class AX(τ) varies over the Kähler moduli and may differ from ΓX at q=1 in some examples. The authors ask for parameter regions where the Γ-class becomes the principal asymptotic class, with special interest in real parameters and an explicit description of such regions.