Cannot pattern match on exist in program definitions

Version

Coq 8.13.2

Operating system

Red Hat Enterprise Linux Workstation 7.4

Description of the problem

It is not possible to pattern match on terms
belonging to the sig type family in the following Program Definitions,
because the witnesses seem to get projected out by some hidden coercion.
However, changing sig into sigT or Ssig will allow
implementing the inspect pattern just fine.

From Coq Require Import Lia Logic.StrictProp NArith.NArith.

Obligation Tactic := idtac.

Section Context.

Import N.

Local Open Scope N_scope.

Context (f : N -> N) (mono : forall (a b : N) (l : a < b), f a < f b).

Print Coercions.

Fail Program Definition f_inspect (n : N) : positive :=
  let m := f (succ n) - f n in
  match exist _ m (eq_refl m) with
  | exist _ N0 e => _
  | exist _ (Npos p) e => p
  end.

Fail Program Definition f_inspect (n : N) : positive :=
  let m := f (succ n) - f n in
  match exist _ m (eq_refl m) in sig _ return positive with
  | exist _ N0 e => _
  | exist _ (Npos p) e => p
  end.

Program Definition f_inspect (n : N) : positive :=
  let m := f (succ n) - f n in
  match existT _ m (eq_refl m) in sigT _ return positive with
  | existT _ N0 e => _
  | existT _ (Npos p) e => p
  end.
Next Obligation. intros n m _ _ e. pose proof mono n (succ n) as l. lia. Qed.

Program Definition f_inspect' (n : N) : positive :=
  let m := f (succ n) - f n in
  match Sexists (fun n : N => Squash (m = n)) m (squash (eq_refl m)) in
  Ssig _ return positive with
  | Sexists _ N0 e => _
  | Sexists _ (Npos p) e => p
  end.
Next Obligation.
  intros n m _ _ e'.
  exfalso. apply sEmpty_ind. induction e' as [e].
  pose proof mono n (succ n) as l. lia. Qed.

End Context.

The command has indeed failed with message:
Found a constructor of inductive type sig while a constructor of N is expected.
The command has indeed failed with message:
Wrong inductive type.

Notes

This may be related to coq/coq#4341 or coq/coq#10877.

Reproduce the provided Coq 8.13.2 example and compare the failing sig matches with the working sigT and Ssig versions. Read the related issues coq/coq#4341 and coq/coq#10877; done means the sig pattern matches in both shown Program Definitions without the reported errors.