Classification conjecture for complete partitions

Let $\Gamma=\Gamma_1\cup\Gamma_2\cup\Gamma_3$ be the family of partitions defined in the paper, and call a partition complete when it has the completeness property defined there.

Complete-partition classification conjecture. A partition $\lambda$ is complete if and only if

$$\lambda\in\Gamma.$$

The paper proves partial classification results and reports exhaustive numerical evidence for the associated characteristic- and score-sequence counts. The source gives no resolution of the full if-and-only-if statement.