Let
$\Theta =(\theta _{j,k})_{3\times 3}$ be a nondegenerate real skew-symmetric
$3\times 3$ matrix, where
$\theta _{j,k}\in [0,1).$ For any
$\varepsilon>0$, we prove that there exists
$\delta>0$ satisfying the following: if
$v_1,v_2,v_3$ are three unitaries in any unital simple separable
$C^*$-algebra A with tracial rank at most one, such that $$\begin{align*}\|v_kv_j-e^{2\pi i \theta_{j,k}}v_jv_k\|<\delta \,\,\,\, \mbox{and}\,\,\,\, \frac{1}{2\pi i}\tau(\log_{\theta}(v_kv_jv_k^*v_j^*))=\theta_{j,k}\end{align*}$$
for all
$\tau \in T(A)$ and
$j,k=1,2,3,$ where
$\log _{\theta }$ is a continuous branch of logarithm (see Definition 4.13) for some real number
$\theta \in [0, 1)$, then there exists a triple of unitaries
$\tilde {v}_1,\tilde {v}_2,\tilde {v}_3\in A$ such that $$\begin{align*}\tilde{v}_k\tilde{v}_j=e^{2\pi i\theta_{j,k} }\tilde{v}_j\tilde{v}_k\,\,\,\,\mbox{and}\,\,\,\,\|\tilde{v}_j-v_j\|<\varepsilon,\,\,j,k=1,2,3.\end{align*}$$
The same conclusion holds if
$\Theta $ is rational or nondegenerate and A is a nuclear purely infinite simple
$C^*$-algebra (where the trace condition is vacuous).
If
$\Theta $ is degenerate and A has tracial rank at most one or is nuclear purely infinite simple, we provide some additional injectivity conditions to get the above conclusion.