By making use of postulated commutation relations of a $\mathrm{CP}$-violating Hamiltonian with the vector and axial-vector charges of chiral SU(2) × SU(2) symmetry, we obtain a theoretical prediction of asymmetry in the slope parameters of ${\tau }^{\pm }$ decay in terms of the $K_L\to 2{\pi }^0$ parameter ${\epsilon }^{\prime }$. We expect $\Delta =\frac{(a^{+}-a^{-})}{(a^{+}+a^{-})}\sim 4{\epsilon }^{\prime }$, where $a^{\pm }$ are the slope parameters in the linear matrix-element approximation of ${\tau }^{\pm }$ decay. With the present experimental limit, $\left|{\epsilon }^{\prime }\right|\lesssim \frac{1}{5}\left|\epsilon \right|$, this gives an upper limit for $\Delta$ of ${10}^{-3\mathrm{}}$.

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