  For $\mathrm{GL}_n$ over a $p$-adic field, Cunningham and Ray proved Vogan's
conjecture, that is, local Arthur packets are the same as ABV packets. They
used the endoscopic theory to reduce the general case to a combinatorial lemma
for irreducible local Arthur parameters, and their proof implies that one can
also prove Vogan's conjecture for $p$-adic $\mathrm{GL}_n$ by proving a
generalized version of this combinatorial lemma. Riddlesden recently proved
this generalized lemma. In this paper, we give a new proof of it, which has its
own interest.
