  Let $X$ be a four-dimensional projective variety defined over the field of
complex numbers with only terminal singularities. We prove that if the
intersection number of the canonical divisor $K$ with every very general curve
is positive ($K$ is almost numerically positive) then every very general proper
subvariety of $X$ is of general type in the viewpoint of geometric Kodaira
dimension. We note that the converse does not hold for simple abelian
varieties.
