Finiteness of all rational and elliptic curves on surfaces of general type

Determine whether every smooth projective surface of general type contains only finitely many rational curves and elliptic curves, including singular ones.

Background

The paper uses a theorem of Lu and Miyaoka asserting finiteness only for smooth rational and smooth elliptic curves on smooth projective surfaces of general type. It explicitly distinguishes this established result from the stronger finiteness question for all rational and elliptic curves, which would include singular curves.