Stability and dimension—a counterexample to a conjecture of Chogoshvili

For every $n \geq 2$ we construct an $n$-dimensional compact subset $X$ of some Euclidean space $E$ so that none of the canonical projections of $E$ on its two-dimensional coordinate subspaces has a stable value when restricted to $X$. This refutes a longstanding claim due to Chogoshvili. To obtain this we study the lattice of upper semicontinuous decompositions of $X$ and in particular its sublattice that consists of monotone decompositions when $X$ is hereditarily indecomposable.