Uniqueness conjecture for the parameter-dependent canonical trace

Let $M$ be a compact manifold and let $\Gamma\subset\mathbb{R}^p$ be a connected cone with nonempty interior and $H^1_{\rm dR}(\Gamma)=0$. Let $\tau:{\rm CL}^*(M,\Gamma)\rightarrow \widetilde{\operatorname{sym}}^*(\Gamma)/\mathcal{P}$ be a linear map satisfying properties (i), (ii), and (iv) of Theorem 3.4. Uniqueness conjecture. There is a constant $c$ such that

$$\tau=c\,\operatorname{TR}.$$

The conjecture asserts that properties (i), (ii), and (iv) uniquely determine the trace $\operatorname{TR}$ up to a scalar factor, without requiring its prescribed values on the larger class of operators in property (iii).