Quantization of the degree of maps from the sphere to the two-sphere

Let $X_1,X_2,X_3\in\operatorname{Mat}(N\times N,\mathbb{C})$ be self-adjoint operators depending on $N$ and satisfying

$$X_1^2+X_2^2+X_3^2=\mathds{1}_{N\times N}.$$

Assume that there is a constant $c$ such that

$$\|X_iX_j-X_jX_i\|\leq\frac{c}{N}$$

for $i,j\in\{1,2,3\}$. Degree-quantization conjecture. Then

$$\operatorname{Tr}\bigl(X_1[X_2,X_3]\bigr)=\frac{2i}{3}k+o(1),$$

where $k\in\mathbb{Z}$ and $|k|\leq \frac{3}{2}c\bigl(1+O(1/N)\bigr)$. This is the matrix analogue of the integer degree of a map from a symplectic surface to $S^2$, suggesting that sufficiently commuting quantum-sphere coordinates retain a quantized topological invariant. The source gives no resolution.