Refined Lefschetz formula with continuous-spectrum multiplicities

Let $\mathfrak a_{\mathbb R}^{*,+}$ be the positive dual cone, let $\overline{\mathfrak a_{\mathbb R}^{*,+}}$ be its closure, and let $\mathbf 1_{\overline{\mathfrak a_{\mathbb R}^{*,+}}}$ denote its indicator function. Let $c(\chi,\eta,{\cal P},\nu,\pi)\in\mathbb Z$ be the coefficients from the integral-trace conjecture. Refined Lefschetz formula. The Lefschetz formula holds with

$$N_{\Gamma,\rm cont}(\pi,\lambda)=\mathbf 1_{\overline{\mathfrak a_{\mathbb R}^{*,+}}}(\lambda)\sum_{\chi,\eta,{\cal P},\nu}c(\chi,\eta,{\cal P},\nu,\pi).$$

This would identify the continuous-spectrum correction multiplicities in the Lefschetz formula with the integer trace coefficients arising from Arthur's distributions. The source gives no proof or resolution of this refinement.