Carrell's minimality conjecture for general-type varieties

Let $X$ be a smooth projective variety of general type. A minimal variety is one with no exceptional $(-1)$-curves in the surface case, and more generally with the minimal-model property. Carrell's conjecture. If $X$ admits a nowhere vanishing holomorphic one form, then $X$ is minimal. This is known for surfaces and threefolds, but remains open in higher dimensions.