Is it possible to unmerge Piecewise expressions?

I have an expression involving Piecewise that I want to break into specific cases. If I let Mathematica's simplification rules loose then it doesn't seem to pick up an obvious cancellation which can be simplified. There doesn't seem to be a builtin way to do this so I'm left with awkward expression rewriting as the only solution. For the specific case I'm interested, see below (update: text code).

IdxAssump=m ∈ Integers && m>= 0 && n ∈ Integers && n >= 0;

p = Piecewise[{{E^(-α Conjugate[α]) Abs[α]^(-2 m + 2 n) n! Hypergeometric1F1Regularized[-m, 1 - m + n, Abs[α]^2]^2/m!, m < n}},
E^(-α Conjugate[α]) Abs[α]^(2 (m - n)) m! Hypergeometric1F1Regularized[-n, 1 + m - n, Abs[α]^2]^2/ n!]

FullSimplify[p /. {α -> 0}, IdxAssump]

p2=Piecewise[Append[p[],{FullSimplify[p[]/.{m->n}],m==n}],p[]]

FullSimplify[p2/.{α->0},IdxAssump] So the question is, can the automatic merging of conditions be picked apart?

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• I have updated the question with copyable input text. I found no help on how to make properly formatted output text so I just put an updated image. – Austin May 8 at 4:45