Boutet de Monvel's conjecture on complexified Hamiltonian flows and Poisson kernels

Let $M$ be a real-analytic manifold with complexification $M_\mathbb{C}$, let $d>0$, and let $P\in \Psi_{\operatorname{phg}}^d(M)$ be analytic with positive classical principal symbol $p$. Assume that $P$ is formally self-adjoint and elliptic, that $p|_{T^*M\setminus 0}>0$, and that the sets

$$\{\xi\in T_x^*M\mid p(x,\xi)^{1/d}\leq 1\}$$

are strictly convex for all $x\in M$. Let $\varphi_t$ be the homogeneous Hamiltonian flow generated by $p^{1/d}$, extended holomorphically for sufficiently small complex $t$, and let $P_\epsilon$ be the Schwartz kernel of $e^{-\epsilon P^{1/d}}$. Define

$$\Phi_x(\xi)=(\pi_x\varphi_{i p(x,\xi)^{1/d}})(x,\xi),$$

for $(x,\xi)\in B_\epsilon^*M\setminus 0$, where

$$B_\epsilon^*M=\{(x,\xi)\in T^*M\mid p(x,\xi)^{1/d}<\epsilon\}.$$

Boutet de Monvel's conjecture. There is a maximal $\epsilon_0>0$ such that, for every $\epsilon\in(0,\epsilon_0)$, the extended flows define a real-analytic diffeomorphism

$$\Phi:B_\epsilon^*M\setminus 0\longrightarrow M_\epsilon^\Phi\setminus M\subset M_\mathbb{C},\qquad (x,\xi)\longmapsto\Phi_x(\xi),$$

where $M_\epsilon^\Phi$ is an open strictly pseudoconvex set in $M_\mathbb{C}$ with orientable $C^\omega$ boundary $\partial M_\epsilon^\Phi$. Moreover, there is a maximal $\epsilon_0'\in(0,\epsilon_0]$ such that, for every $\epsilon\in(0,\epsilon_0')$, the map $x\mapsto P_\epsilon(x,y)$ extends holomorphically to $M_\epsilon^\Phi$ for each fixed $y\in M$, the restriction $P_\epsilon|_{\partial M_\epsilon^\Phi\times M}$ induces a complex-phase PHG Fourier integral operator $S_\epsilon$ of order $-(n-1)/4$, and

$$S_\epsilon:H^s(M)\longrightarrow\mathcal{O}^{s+(n-1)/4}(\partial M_\epsilon^\Phi)$$

is a homeomorphism for every $s\in\mathbb{R}$. The strict pseudoconvexity also implies that $M_\epsilon^\Phi$ admits a Kähler metric with a global potential. This conjecture generalizes the result known for $P=\sqrt{-\Delta_g}$; its general validity was stated by Boutet de Monvel but remains unproved.