Huang–Ji–Yin Gap Conjecture for CR mappings between spheres

For $n\geq 2$, let $\kappa_0$ be the largest integer such that

$$(\kappa-1)n+\kappa\leq \sum_{i=0}^{\kappa-1}(n-i)-1,$$

and define

$$I_\kappa=\{j\in\mathbb N:(\kappa-1)n+\kappa\leq j\leq \sum_{i=0}^{\kappa-1}(n-i)-1\},\qquad \kappa=1,\ldots,\kappa_0.$$

Let $f\colon\mathbb S^n\to\mathbb S^N$ be a sufficiently smooth CR mapping. Huang–Ji–Yin Gap Conjecture. If $N-n\in I_\kappa$, then there exists an integer $n\leq N_0<N$ with

$$N_0-n\leq (\kappa-1)n-\kappa-1$$

and an affine complex subspace $A^{N_0+1}$ of dimension $N_0+1$ such that $f(\mathbb S^n)\subset S^N\cap A^{N_0+1}$. This predicts rigidity of CR mappings in the specified codimension gaps, but the source presents it as an unresolved conjecture and uses it as the geometric statement implied by the SOS conjecture.