The complete-list conjecture for Ulrich partition types

Let $(\alpha,\beta,\gamma)$ be the type of an Ulrich partition, and call two types symmetric when they are identified by the symmetry used in the paper. Here $n$ is a positive integer, and each $k_i$ has the form

$$k_i=\frac{4^{m_i+1}-1}{3}$$

for some $m_i\geq0$. Complete-list conjecture. Up to symmetry, the type $(\alpha,\beta,\gamma)$ is one of

$$(1,n,1),\quad (1,n,2),\quad (2,2n,2),\quad (2,2,3),\quad (2,1,k_1),\quad (1,2,k_1),\quad (1,2,k_1+k_2).$$

The conjecture proposes that the examples obtained in the paper exhaust all Ulrich partition types for two-step flag varieties. The paper gives partial classifications and states that the classification remains incomplete, so the conjecture is open.