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In [C. Palamidessi, V. Saraswat, F. Valencia and B. Victor. On the Expressiveness of Linearity vs Persistence in the Asynchronous Pi Calculus. LICS 2006:59-68, 2006] the authors studied the expressiveness of persistence in the asynchronous @p-calculus (A@p) wrt weak barbed congruence. The study is incomplete because it ignores the issue of divergence. In this paper, we present an expressiveness study of persistence in the asynchronous @p-calculus (A@p) wrt De Nicola and Hennessy's testing scenario which is sensitive to divergence. Following [C. Palamidessi, V. Saraswat, F. Valencia and B. Victor. On the Expressiveness of Linearity vs Persistence in the Asynchronous Pi Calculus. LICS 2006:59-68, 2006], we consider A@p and three sub-languages of it, each capturing one source of persistence: the persistent-input calculus (PIA@p), the persistent-output calculus (POA@p) and persistent calculus (PA@p). In [C. Palamidessi, V. Saraswat, F. Valencia and B. Victor. On the Expressiveness of Linearity vs Persistence in the Asynchronous Pi Calculus. LICS 2006:59-68, 2006] the authors showed encodings from A@p into the semi-persistent calculi (i.e., POA@p and PIA@p) correct wrt weak barbed congruence. In this paper we prove that, under some general conditions, there cannot be an encoding from A@p into a (semi)-persistent calculus preserving the must testing semantics.