Thread-set conjecture for iterated localizations of stratified tensor triangulated categories

Let ${\mathcal{T}}$ be a rigidly-compactly generated tensor triangulated category, stratified in the sense of Balmer–Favi, whose Balmer spectrum $\operatorname{Spc}({\mathcal{T}}^c)$ is noetherian and has finite Krull dimension. For a tuple ${\mathbb{A}}=(A_1,\dots,A_k)$ of subsets of $\operatorname{Spc}({\mathcal{T}}^c)$, a thread set is a chain $T\subseteq\operatorname{Spc}({\mathcal{T}}^c)$ containing elements $a_i\in T\cap A_i$ with $a_1\supseteq\cdots\supseteq a_k$. Let ${\mathbb{L}}_{\mathbb{A}}$ denote the associated iterated localization.

Thread-set conjecture. Let ${\mathbb{A}}$ and ${\mathbb{B}}$ be tuples of subsets of $\operatorname{Spc}({\mathcal{T}}^c)$. If they have the same thread sets, then there is a canonical natural isomorphism

$${\mathbb{L}}_{\mathbb{A}}\cong {\mathbb{L}}_{\mathbb{B}}.$$

The conjecture proposes that the composition of localizations is determined by the combinatorics of chains of Balmer primes in the indexing subsets, and hence by the Balmer spectrum. Its status is not resolved in the supplied source.