Smoothness conjecture for the singular part of the distributional leafwise trace

Let $\mathcal{F}$ be a foliation with dense leaves, and write $\operatorname{codim}\mathcal{F}$ for its codimension. For each degree $i$, let $\operatorname{Tr}_{\mathrm{dis}}^i(\mathcal{F})$ be the distributional trace and let $\beta^i_\Lambda(\mathcal{F})$ be the corresponding $\Lambda$-Betti number; denote by $\delta_e$ the delta distribution at the identity element $e$.

The smoothness conjecture. If $\operatorname{codim}\mathcal{F}>0$ and the leaves are dense, then

$$\operatorname{Tr}_{\mathrm{dis}}^i(\mathcal{F})-\beta^i_\Lambda(\mathcal{F})\cdot\delta_e$$

is $C^\infty$ around $e$ for each degree $i$.

This generalizes the preceding result in dimension two, where the difference between the distributional trace in degree one and its singular delta contribution is smooth near the identity. The source does not provide a resolution, so the conjecture remains open.