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In this paper, we study the structure of the set of local Arthur parameterssuch that the corresponding local Arthur packets all contain a givenrepresentation of the split classical groups $G_n=Sp_{2n}, SO_{2n+1}$ over anon-Archimedean local field of characteristic zero. We show that there exist apair of elements in this set which are the unique maximal and minimal elementsunder various orderings. As an application, we prove the closure orderingconjecture on the local $L$-parameters of representations in local Arthurpackets of $G_n$, namely, given any representation $\pi$ in a local Arthurpacket $\Pi_{\psi}$, the closure of the local $L$-parameter of $\pi$ in theVogan variety must contain the local $L$-parameter corresponding to $\psi$.This conjecture is inspired by the work of Adams, Barbasch, and Vogan, andCunningham et al. on ABV-packets. Its ABV-analogue has been proved byCunningham et al. Our result provides evidence for the coincidence of the localArthur packets with the ABV-packets as conjectured by Cunningham et al. in[CFMMX22, Section 8.3, Conjecture 1]. Our key ingredients to overcome thedifficulty which lies in the nontrivial intersections of local Arthur packetsare the operators developed in [HLL22] and a reduction to the unique maximalelement above. The closure ordering conjecture reveals a very interestinggeometric nature of local Arthur packets and is expected to have many importantimplications. We show that it implies the Enhanced Shahidi's Conjecture forgeneral quasi-split connected reductive groups, under certain reasonableassumptions. We verify these assumptions for $G_n$, giving a new proof of theEnhanced Shahidi's Conjecture. As a byproduct of the proof of the closureordering conjecture, we give new algorithms to determine whether a givenrepresentation is of Arthur type, which have their own interests andtheoretical importance.

The closure ordering conjecture on local $L$-parameters in local Arthur packets of classical groups