Does the doctrine of fruit of the poisonous tree doctrine apply to the Miranda exclusionary rule?

The case relied on by my colleagues that leads them to this doubtful conclusion is Oregon v. Elstad, 470 U.S. 298 (1985). Elstad declined to apply the Fourth Amendment fruit of the poisonous tree doctrine to Miranda violations. 470 U.S. at 309 ("If errors are made . . . in administering the prophylactic Miranda procedures, they should not breed the same irremediable consequences as police infringement of the Fifth Amendment itself."); id. at 306 ("The Miranda exclusionary rule . . . sweeps more broadly than the Fifth Amendment. . . ."); id. ("[A] procedural Miranda violation differs in significant respects from violation of the Fourth Amendment. . . ."); id.