Universal recursive-polynomial conjecture for Fano hypersurfaces

Let $M_N^k$ be a Fano hypersurface, and let $L_m^{N,k,d}$ be its structure constants for rational curves of degree $d$. For each $d$, consider the structure constants $L_m^{N+1,k,n}$ with $1\leq n\leq d$, and let $\gamma_0^{N,k,d}$ denote the coefficient in the fundamental relation used in the paper.

Universal recursion conjecture. There are universal recursive polynomial laws expressing

$$L_m^{N,k,d}$$

in terms of

$$L_m^{N+1,k,n},\qquad 1\leq n\leq d.$$

The recursive polynomials are invariant when $N-k\geq2$, and imply

$$\gamma_0^{N,k,d}=\gamma_0^{N+1,k,d}.$$

When $N-k=1$, the recursive formulas change only for lines, namely $d=1$.

These laws are proposed as the general mechanism for descending recursively in $N$. The source explicitly describes them as conjectural and later uses them to propose formulas for higher-degree curves; no proof or resolution is given.