The nonexistence conjecture for Ulrich partitions with two large blocks

Let $(\alpha,\beta,\gamma)$ be the type of an Ulrich partition. Nonexistence conjecture. There are no Ulrich partitions of type $(\alpha,\beta,\gamma)$ if $\beta\geq3$ and $\gamma\geq3$. The paper states that its partial classification implies this is equivalent to the complete-list conjecture above; since that classification is incomplete, the nonexistence statement remains open.