The arithmetic intersection conjecture for arbitrary fundamental matrices

Let ${\mathcal N}_n$ be the relevant unitary Rapoport–Zink space, and let ${\mathcal Z}(x_1),\ldots,{\mathcal Z}(x_n)$ be special divisors whose fundamental matrix is equivalent to

$$\operatorname{diag}(\pi^{r_1},\pi^{r_2},\ldots,\pi^{r_n}),\qquad r_1\geq r_2\geq\cdots\geq r_n.$$

The annihilation conjecture. The element $\pi^{r_1}$ annihilates the structure sheaf of

$${\mathcal Z}(x_1)\cap{\mathcal Z}(x_2)\cap\cdots\cap{\mathcal Z}(x_n).$$

The paper proves the assertion in the special minuscule case, where the exponents have the form $(m,0,\ldots,0)$, and presents that result as evidence for the general conjecture. The arbitrary-exponent statement remains open in the source.