Schapira–Schneiders conjecture on the Euler class of a holonomic complex

Let $X$ be a complex manifold of dimension $d$, let ${\cal M}^\bullet$ be an object of $\operatorname{D}^b_{good}({\cal D}^{op}_X)$, and let $\Lambda$ be a conic subvariety of $T^*X$ containing $\operatorname{char}({\cal M}^\bullet)$. The symbol complex is $\sigma({\cal M}^\bullet)=\pi^{-1}gr{\cal M}^\bullet\otimes_{\pi^{-1}gr{\cal D}_X}{\cal O}_{T^*X}$, where $\pi:T^*X\to X$ is the projection, and $\mu\operatorname{eu}({\cal M}^\bullet)$ denotes its microlocal Euler class. Schapira–Schneiders' conjecture.

$$\mu\operatorname{eu}({\cal M}^\bullet)=\left[ch_\Lambda\left(\sigma({\cal M}^\bullet)\smile\pi^*Td(TX)\right)\right]^{2d}.$$

Here $ch_\Lambda$ is the Chern character with support in $\Lambda$, and $[\,\cdot\,]^{2d}$ denotes the homogeneous component of degree $2d$. This conjecture identifies the microlocal Euler class of a good $\mathcal D_X$-complex with a characteristic class of its symbol, providing the Riemann–Roch-type formula underlying the index theorem for elliptic pairs.