Forward stability conjecture for the Schur canonical form

Let $A_0=U_0T_0U_0^*\in\mathbb{C}^{n\times n}$, where $U_0$ is unitary and $T_0$ is upper triangular. A factorization of $A$ as $A=UTU^*$ is a Schur decomposition when $U$ is unitary and $T$ is upper triangular. Forward stability conjecture. There exist constants $K,\epsilon>0$, depending only on $A_0$, such that for every $A$ satisfying

$$\|A-A_0\|<\epsilon,$$

there is a Schur decomposition $A=UTU^*$ for which

$$\|U-U_0\|+\|T-T_0\|\leq K\|A-A_0\|^{1/n}.$$

The conjecture seeks a Hölder-type forward stability bound for the Schur factors, with the exponent $1/n$ matching the general eigenvalue stability bound. Whether such a factorization always exists with constants depending only on $A_0$ is not resolved by the supplied context.