The kind of task to be carried out in this talk can be seen as belonging to, quoting Kreisel, “the sort of Kleinarbeit which is generally needed to support a genuine hypothesis (...) as opposed to a mere mathematical fancy.” (Kreisel 1971, p.114). We propose a brief analysis of an argumentative strategy to justify the completeness claim of the so-called normalisation thesis on identity of proofs (cf. Prawitz 1971), attributed to Barendregt by Kreisel in Kreisel 1971, based on proving the maximality of the equivalence relation corresponding to the thesis. This strategy became of particular significance after the actual obtainment of the maximality results of e.g. Widebäck 2001 and Došen and Petrić 2000 and 2001. Resorting to taxonomical remarks on criteria for identity of proofs and observing some related formal results discussed in de Castro Alves 2019, we argue that literature has endorsed an unfairly favourable impression of the degree of effectiveness of the justification for the normalisation thesis that such maximality results are capable of yielding as premisses in Barendregt’s argumentative strategy.