Uniqueness of rectifiers up to equivalence

Let $T$ be a $K$-minisotropic torus in $G$ that splits over an unramified extension $L$. A rectifier for $T$ is a function

$$\boldsymbol\mu: (T, \xi) \mapsto \mu_{\xi}$$

that attaches to each $(T,\xi)\in P_G(K)$ a character $\mu_{\xi}$ of $T(L)_{\Gamma}$ satisfying the defining tameness, descent, local Langlands, and compatibility conditions; two rectifiers are equivalent if they differ by some $\alpha\in Q_T$ on depth-zero characters and agree on positive-depth characters.

Uniqueness of rectifiers. For $T$ as above, $T$ admits a unique rectifier up to equivalence.

Rectifiers correct the character attached to a Langlands parameter so that it descends from $T(L)_{\Gamma}$ to a character of $T(K)$ and realizes the local Langlands correspondence. The claim is presented in the paper as the uniqueness result underlying its treatment of the unramified case; the supplied text gives no resolution status beyond the assertion itself.