Dynamical Lefschetz trace formula for foliated spaces

Let $X$ be a compact manifold with a one-codimensional foliation ${\mathcal F}$ and an ${\mathcal F}$-compatible flow $\phi$. Assume that the fixed points and periodic orbits are non-degenerate as described: fixed-point tangent maps have no eigenvalue $1$ for $t>0$, and for every closed orbit $\gamma$ of length $l(\gamma)$, every $x\in\gamma$, and every nonzero integer $k$, $T_x\phi^{kl(\gamma)}$ has eigenvalue $1$ with algebraic multiplicity one. Let $\bar{H}^n_{\mathcal F}(X)$ be the reduced leafwise cohomology and let ${\mathcal D}'(J)$ denote Schwartz distributions on an open subset $J\subset\mathbb R$. For a fixed point $x$, set $\varepsilon_x=\operatorname{sgn}\det(1-T_x\phi^t\mid T_x{\mathcal F})$. For a closed orbit $\gamma$ and $k\in\mathbb Z\setminus\{0\}$, set $\varepsilon_\gamma(k)=\operatorname{sgn}\det(1-T_x\phi^{kl(\gamma)}\mid T_xX/\mathbb RY_{\phi,x})$. Dynamical Lefschetz trace formula. There should exist a natural definition of a ${\mathcal D}'(\mathbb R^{>0})$-valued trace of $\phi^*$ on $\bar{H}^{\raisebox{0.05cm}{\scriptscriptstyle\bullet}}_{\mathcal F}(X)$ such that, in ${\mathcal D}'(\mathbb R^{>0})$,

$$\sum_{n=0}^{\dim\mathcal F}(-1)^n\operatorname{Tr}(\phi^*\mid\bar{H}^n_{\mathcal F}(X))=\sum_\gamma l(\gamma)\sum_{k=1}^{\infty}\varepsilon_\gamma(k)\delta_{kl(\gamma)}+\sum_x\varepsilon_x|1-e^{\kappa_xt}|^{-1}.$$

This conjectural formula relates compact flow orbits and fixed points to the alternating trace on reduced leafwise cohomology; it was motivated by a non-rigorous argument of Guillemin, later rediscovered by Patterson. The supplied text does not establish the claimed natural trace or the formula.