Lower-bound conjecture for irreducible $\lambda$-quidditiy length in characteristic two

Let $\mathbb{K}$ be a finite field of characteristic $2$, let $\operatorname{card}(\mathbb{K})$ denote its cardinality, and let $\ell_{\mathbb{K}}$ denote the invariant defined in the paper for $\mathbb{K}$. Characteristic-two lower-bound conjecture. One has

$$\ell_{\mathbb{K}}\geq 3\bigl(\operatorname{card}(\mathbb{K})-1\bigr).$$

This conjecture concerns the general lower bound in characteristic two. The source does not provide a proof or resolution.