Ulrich's conjecture on bundles for smooth projective varieties

Let $X\subset \mathbb{P}^{N}$ be a smooth projective variety, and call a vector bundle on $X$ Ulrich when it satisfies the Ulrich conditions for the given embedding. Ulrich's conjecture. Every smooth projective variety $X\subset \mathbb{P}^{N}$ has an Ulrich bundle. The conjecture is described as an optimistic expectation; Ulrich bundles have been constructed for many specific varieties, but the general existence problem remains open.