  We prove the Kawaguchi-Silverman conjecture (KSC), about the equality of
arithmetic degree and dynamical degree, for every surjective endomorphism of
any (possibly singular) projective surface. In high dimensions, we show that
KSC holds for every surjective endomorphism of any $\mathbb{Q}$-factorial
Kawamata log terminal projective variety admitting an int-amplified
endomorphism, provided that KSC holds for any surjective endomorphism with the
ramification divisor being totally invariant and irreducible. In particular, we
show that KSC holds for every surjective endomorphism of any rationally
connected smooth projective threefold admitting an int-amplified endomorphism.
The main ingredients are the equivariant minimal model program, the
effectiveness of the anti-canonical divisor and a characterization of toric
pairs.
