Hjorth–Kechris–Louveau strictness conjecture for Borel equivalence-relation hierarchies

For $n\geq 3$ and $0\leq k\leq n-2$, let $\cong^\ast_{n,k}$ be the equivalence relation whose invariants are pairs $(A,R)$ with $A$ a hereditarily countable set in $\mathcal{P}^n(\mathbb{N})$ and, for each $a\in A$, $R(a,-,-)$ an injective map from $A$ into $\mathcal{P}^k(\mathbb{N})$. For a limit ordinal $\lambda$, define $\cong^\ast_{\lambda+1,\beta}$ and $\cong^\ast_{\lambda+n,\beta}$ analogously.

Hjorth–Kechris–Louveau strictness conjecture. The following Borel reducibilities are strict:

$$\cong^\ast_{n,l}<_B\cong^\ast_{n,k}$$

for any $n\geq 3$ and $l<k\leq n-2$;

$$\cong^\ast_{\lambda+1,\alpha}<_B\cong^\ast_{\lambda+1,\beta}$$

for limit $\lambda$ and $\alpha<\beta<\lambda$; and

$$\cong^\ast_{\lambda+n,\alpha}<_B\cong^\ast_{\lambda+n,\beta}$$

for limit $\lambda$, $n\geq 2$, and $\alpha<\beta\leq\lambda+n-2$.

These conjectures concern whether the invariant hierarchies introduced by Hjorth, Kechris and Louveau collapse under Borel reducibility. The surrounding results establish maximality at several Borel complexity levels and stronger reducibility bounds for equivalence relations induced by abelian closed subgroups, but do not settle strictness.