Automatic LIIA Independent of Locking Order

Ranked Pairs satisfy Local Independence of Irrelevant Alternatives criterion,
but I got curious if this property is obtained independently of locking order.

For context, ISDA comes independently of locking order,
but ISDA is implied by LIIA + Majority criterion,
so I got curious if LIIA is what actually is obtained
and ISDA simply followed from it.

I tried to create a proof for a positive result,
but quickly discovered I could not figure out how
to cover scenarios containing multiple matchups
to be locked in at the same time.

Any help would be much appreciated.
Gustav

P.S: I have started to suspect I need to fail LIIA
for a MMPO locking order to satisfy all of
AFB+Mono+LN-Harm+MB-ISDA
unless LIIA satisfaction is automatic independently of locking order,
and then figured it was interesting enough of a question on its own.

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