Modulo-4 asymptotics for commutators of spectral projections of spin operators

Let $J_x,J_y,J_z$ be the generators of an irreducible, unitary, $n$-dimensional representation of $SU(2)$, and set

$$C_n=\left[\mathbb{1}_{(0,\infty)}(J_x),\mathbb{1}_{(0,\infty)}(J_z)\right].$$

Write $o(1)$ for a quantity tending to zero as $n\to\infty$. The modulo-4 asymptotic conjecture. The operator norms satisfy

$$\frac{\lVert C_{4n+3}\rVert_{\operatorname{op}}-\lVert C_{4n+1}\rVert_{\operatorname{op}}}{\lVert C_{4n+3}\rVert_{\operatorname{op}}-\lVert C_{4n}\rVert_{\operatorname{op}}}=o(1)$$

and

$$\frac{\lVert C_{4n+3}\rVert_{\operatorname{op}}-\lVert C_{4n+1}\rVert_{\operatorname{op}}}{\frac 1 2-\lVert C_{4n+p}\rVert_{\operatorname{op}}}=o(1)$$

for $p=0,1,3$. These asymptotics describe the apparent dependence of the norms on the representation dimension modulo $4$, complementing the proved facts that $\lVert C_{4n+2}\rVert_{\operatorname{op}}=\frac12$ and $\lim_{n\to\infty}\lVert C_n\rVert_{\operatorname{op}}=\frac12$.