  A finite constraint language $\mathscr{R}$ is a finite set of relations over
some finite domain $A$. We show that intractability of the constraint
satisfaction problem $\operatorname{CSP}(\mathscr{R})$ can, in all known cases,
be replaced by an infinite hierarchy of intractable promise problems of
increasingly disparate promise conditions: where instances are guaranteed to
either have no solutions at all, or to be $k$-robustly satisfiable (for any
fixed $k$), meaning that every "reasonable" partial instantiation on~$k$
variables extends to a solution. For example, subject to the assumption
$\texttt{P}\neq \texttt{NP}$, then for any~$k$, we show that there is no
polynomial time algorithm that can distinguish non-$3$-colourable graphs, from
those for which any reasonable $3$-colouring of any $k$ of the vertices can
extend to a full $3$-colouring. Our main result shows that an analogous
statement holds for all known intractable constraint problems over fixed finite
constraint languages.
