Equivariant Ulrich bundles on exceptional homogeneous varieties

We prove that the only rational homogeneous varieties with Picard number 1 of the exceptional algebraic groups admitting irreducible equivariant Ulrich vector bundles are the Cayley plane $E_6\mathrm{/}{P}_1$ and the $E_7$-adjoint variety $E_7\mathrm{/}{P}_1$. From this result, we see that a general hyperplane section $F_4\mathrm{/}{P}_4$ of the Cayley plane also has an equivariant but non-irreducible Ulrich bundle.