Dream Building Blocks conjecture for surjective endomorphisms

Let $X$ be a normal projective variety and let $f:X\to X$ be a non-isomorphic surjective endomorphism. The properties weakly $\delta$-imprimitive, strongly imprimitive, quasi-abelian, and polarized are the alternatives under consideration. Dream Building Blocks conjecture. The endomorphism $f$ is either weakly $\delta$-imprimitive, strongly imprimitive, quasi-abelian, or polarized after iteration. This conjecture has been verified for surfaces, even without the strongly imprimitive possibility, and for smooth projective threefolds. Its status in the stated generality remains open.