Universality of the Euclidean form on affine type C admissible tuples

For an integer $n\geq1$, define

$$\Delta_n^C=\left\{(x_1,\ldots,x_n)\in\mathbb{Z}^n\ \middle|\ x_i+i\not\equiv0\pmod{2n+1},\ x_i+i\not\equiv x_j+j\pmod{2n+1},\ x_i+i\not\equiv-(x_j+j)\pmod{2n+1}\right\},$$

where the latter two conditions hold for $1\leq i<j\leq n$. The Euclidean form is $\sum_{i=1}^n x_i^2$. Affine type C universality conjecture. For any integer $n\geq4$, the Euclidean form is universal on $\Delta_n^C$. The claim is proved in the paper when $2n+1$ is prime; the general case remains open.