Grading conjecture for the lambda-degree of the lambda-value algebra

Let $\mathcal{L}(K)$ be the $K$-algebra generated by the lambda values, equipped with the lambda-degree induced by the lambda-type of composition arrays and taking values in the monoid $\boldsymbol{S}$. Lambda-degree grading conjecture. The lambda-degree defines a grading of the algebra $\mathcal{L}(K)$ by the monoid $\boldsymbol{S}$. The source says that this property will be studied in another work and gives no resolution.