The timelike tube theorem states that the additive algebra $A_{\text{add}}(U)$ of operators in a spacetime region $U$ is equal to the additive algebra $A_{\text{add}}(E(U))$ of the timelike envelope $E(U)$ of $U$.
One might hope for a further generalization that $A_{\text{add}}(U) = A_{\text{add}}(U'')$, where $U''$ is the causal completion of $U$, defined as $(U')'$ where $U'$ is the set of points spacelike separated from $U$. The causal completion $U''$ contains the timelike envelope $E(U)$ but can be strictly larger.
However, there are some counterexamples showing that $A_{\text{add}}(U) = A_{\text{add}}(U'')$ does not hold in general quantum field theories. For example, if $U$ is a disconnected union of two balls, one to the future of the other, then in a massless free field theory in even spacetime dimensions, operators in the causal completion $U''$ will not always be contained in the additive algebra $A_{\text{add}}(U)$.
Nevertheless, one might speculate that in a "sufficiently complex" interacting quantum field theory, $A_{\text{add}}(U) = A_{\text{add}}(U'')$ might still hold in general. The idea is that a sufficiently rich set of operators in $U$ could be used to construct probes that could measure any operator in $U''$.
What is known about the status of this conjecture that $$A_{\text{add}}(U) = A_{\text{add}}(U'')$$ in a sufficiently complex QFT? Are there interesting classes of QFTs beyond free field theory where it has been proved or disproved? Or is it an open question deserving further study? It seems like an intriguing extension of the timelike tube theorem that would nicely unify it with ordinary causality encoded by the statement $$A_{\text{add}}(U) = A_{\text{add}}(D(U))$$ where $D(U)$ is the domain of dependence.