Self-$K$-isocorrespondence conjecture for Kodaira dimension

Let $X$ be a projective variety, and let a self-$K$-isocorrespondence be a correspondence $\Sigma\subset X\times X$ satisfying the self-$K$-isocorrespondence condition used in the paper. Let $\kappa(X)$ denote the Kodaira dimension of $X$.

Self-$K$-isocorrespondence conjecture. If there exists a self-$K$-isocorrespondence $\Sigma\subset X\times X$ passing through an arbitrary pair $(x,y)$, then

$$\kappa(X)\leq 0.$$

The source presents this as a converse to examples on $K$-trivial varieties and notes that the maximal Kodaira-dimension case is excluded from the examples; no resolution is supplied.