7 problems

Rapoport–Zink's conjecture. There exists an open subspace

Potentially unramified tower conjecture. The following assertions hold: (1) $\hat{\tilde{X}}$ is an adic space and $(\hat{\tilde{X}})^\diamond=\tilde{X}^\diamond$; (2) the natural…

Comparison-isomorphism conjecture. The morphism $\Upsilon^{\mathrm{sm}}$ is an isomorphism.

Let $T_1\to T_2$ be a localized intrinsic étale morphism of strongly sheafy adic spaces. Local structure conjecture. Any localized intrinsic étale morphism of strongly sheafy adic…

Fargues's conjecture. The Banach algebra $B_I$ is strongly noetherian: for every $n\geq 1$, the Tate algebra

The étale-topos inverse-limit conjecture. The étale topos of $X$ is equivalent to the projective limit of the étale topoi of the $X_i$.

Extension conjecture. The claim of the theorem and the corollary also holds true if one replaces $x_0$ by a (locally) closed subscheme of the special fiber over which there exists…