Semiclassical asymptotics conjecture for scalar products of eigen-half-densities

Let $M$ be a $2n$-dimensional Poisson manifold with a real polarization $P$ and two integrable systems given by Lagrangian fibrations $\pi_i:M\to B_i$, with generically transverse fibers and fibers transverse to the leaves of $P$. Let $C_h(M,B_i)$ be the corresponding commuting quantized subalgebras acting on $H_P^{(1/2)}$, and let $\psi^{(i)}_{b_i}$ be semiclassical eigen-half-densities. For transverse fibers $\mathcal{L}^{(1)}_{b_1}$ and $\mathcal{L}^{(2)}_{b_2}$, choose reference points and paths as in the source, and write $S_{\gamma_1,\gamma_2}=S^{(1)}_{\gamma_1}-S^{(2)}_{\gamma_2}$ for the difference of the corresponding prequantization actions. Semiclassical scalar-product conjecture. The scalar product has the asymptotic structure

$$(\psi^{(2)}_{b_2},\psi^{(1)}_{b_1})=\frac{C}{(2\pi h)^{n/2}}\sum_{c\in\mathcal{L}^{(1)}_{b_1}\cap\mathcal{L}^{(2)}_{b_2}}e^{\frac{i}{h}S_{\gamma_1,\gamma_2}(c,b_1,b_2)+\frac{i\pi}{2}\mu_{\gamma_1,\gamma_2}(c)}\left|\det\left(\frac{\partial^2S_{\gamma_1,\gamma_2}(c,b_1,b_2)}{\partial b_1^i\partial b_2^j}\right)\right|^{1/2}(1+O(h))\sqrt{|db_1db_2|}.$$

Here $\mu_{\gamma_1,\gamma_2}(c)$ is the Maslov index, and under the Bohr–Sommerfeld conditions the phase is independent of the path choices; the Hessian and higher-order terms are independent of the reference points. This is the expected semiclassical stationary-phase formula for the Blattner–Kostant–Sternberg-type kernel, but the supplied text does not establish its status beyond stating the formula.