The optimal filtration bound for rational curves on cubic fourfolds

Let $Y\subset\mathbb{P}^5$ be a nonsingular cubic fourfold, and let $S_\bullet(Y)$ be the filtration on $\mathrm{CH}_1(Y)$ defined using special and canonical lines. For a nonsingular connected rational curve $C\subset Y$, write $e=\deg C$ and define

$$\mathbf{b}(e)= \begin{cases} \frac{3e}{2} & e\text{ even},\\ \frac{3e+1}{2} & e\text{ odd}. \end{cases}$$

Optimal filtration bound conjecture. For every nonsingular connected rational curve $C\subset Y$ of degree $e\geq5$,

$$[C]\in S_{\mathbf{b}(e)}(Y).$$

This is motivated by the proposed relationship between moduli spaces of rational curves, moduli of stable objects in ${\mathcal A}_Y$, and Voisin’s constant-cycle subvariety conjecture. The source presents it as a speculation and gives no proof for all $e\geq5$.