The dimension of automorphism groups of algebraic varieties with pseudo-effective log canonical divisors

Let $(X,D)$ be a log smooth pair of dimension $n$, where $D$ is a reduced effective divisor such that the log canonical divisor $K_X + D$ is pseudo-effective. Let $G$ be a connected algebraic subgroup of $\mathrm{Aut}(X,D)$. We show that $G$ is a semi-abelian variety of dimension $\le \min\{n-\bar{\kappa}(V), n\}$ with $V := X\setminus D$. In the dimension two, Shigeru Iitaka claimed in his 1979 Osaka J. Math. paper that $\dim G\le \bar{q}(V)$ for a log smooth surface pair with $\bar{\kappa}(V) = 0$ and $\bar{p}_g(V) = 1$. We (re)prove and generalize this classical result for all surfaces with $\bar{\kappa}=0$ without assuming Iitaka's classification of logarithmic Iitaka surfaces or logarithmic $K3$ surfaces.