Classification of attainable complexities for arbitrary spaces

For a topological space $X$, let $\operatorname{Compl}(X)$ be its set of attainable complexities. Classification conjecture. Exactly one of the following alternatives holds:

$$\begin{array}{ll} (1)&X\text{ is not }\mathcal K\text{-analytic and }\operatorname{Compl}(X)=\emptyset;\\ (2)&X\text{ is compact and }\operatorname{Compl}(X)=\{0\};\\ (3)&X\text{ is }\sigma\text{-compact and }\operatorname{Compl}(X)=\{1\};\\ (4)&\operatorname{Compl}(X)=[\alpha,\beta]\text{ for some }2\leq\alpha\leq\beta\leq\omega_1. \end{array}$$

Moreover, each of these possibilities occurs for some space $X$. This is the proposed complete solution of the attainable-complexity problem conditional on the closed-interval conjecture above.