A celebrated problem of Hirzebruch aims at the description of all possible smooth compactifications of C^n with b_2=1. Motivated by this problem, we will consider the case where n=4 and study 8-dimensional compact symplectic manifolds with b_2=1 with the additional restriction of b_4=2. If these manifolds admit a Hamiltonian action of a $2$-dimensional torus with isolated fixed points, we will see that their isotropy data, (equivariant) cohomology rings and (equivariant) Chern classes must agree with those of the only known examples of smooth compactifications of C^4 with the prescribed Betti numbers that admit a T^2-action: the quintic del Pezzo 4-fold W_5, the smooth Fano-Mukai 4-fold V_!8 and the Grassmannian of complex 2-planes in C^4. This is joint work with Nicholas Lindsay and Silvia Sabatini.