Deficient-score-sequence characterization of complete partitions

Let $\lambda$ be a partition, let $\lambda'\subseteq\lambda$ denote a subpartition in the paper's sense, and let $\mathfrak{S}^2(\lambda')$ be the set of score sequences of $2$-tournaments with shape $\lambda'$. A sequence is deficient according to the paper's preceding definition.

Deficient-sequence characterization conjecture. A partition $\lambda$ is complete if and only if, for every $\lambda'\subseteq\lambda$, the set $\mathfrak{S}^2(\lambda')$ has no deficient sequences:

$$\mathfrak{S}^2(\lambda')\text{ has no deficient sequences for all }\lambda'\subseteq\lambda.$$

The source states that this is equivalent to the complete-partition classification conjecture. The surrounding theorem gives partial implications, but the source gives no resolution of the conjecture.