The closed-interval conjecture for attainable complexities of mathcal K-analytic spaces

For a space $X$, let $\operatorname{Compl}(X)$ denote its set of attainable complexities. A space is $\mathcal K$-analytic if it has the standard $\mathcal K$-analyticity property used in the paper. Closed-interval conjecture. For every $\mathcal K$-analytic space $X$, the set $\operatorname{Compl}(X)$ is an interval and is closed in $[0,\omega_1]$.

If true, this would, together with the theorem that every closed interval contained in $[2,\omega_1]$ occurs as $\operatorname{Compl}(X)$, give a complete classification of the possible attainable-complexity sets. The claim is presented as a weak conjecture; its resolution would settle the corresponding problem for arbitrary spaces.