Bicategorical classifiers and truncations for partitioned groupoid assemblies

Let $A$ be the partial combinatory algebra underlying the assembly category, and let $\mathrm{p}\mathbbm{Grpd}(\operatorname{Asm}(A))$ denote the category of partitioned groupoid assemblies. For a morphism $f:X\to Y$, write $f_h^*$ for the bifunctor on slices induced by bipullback. Bicategorical classifier conjecture. The following should hold for $\mathrm{p}\mathbbm{Grpd}(\operatorname{Asm}(A))$: bipullback along every $f$ has both a left and a right biadjoint; there is a sub-object biclassifier classifying $-1$-truncated morphisms; for every Grothendieck universe $V$, there is an object biclassifier classifying $V$-fibred $0$-truncated morphisms; and there is a biclassifier for modest $0$-truncated morphisms, with the class of such morphisms closed under bipushforward. The paper notes that the notions of $n$-truncated and modest morphism still need to be developed, and additionally suggests a univalence condition for the two object biclassifiers.