Continuing the investigation initiated here, and applying the same notation.

Proposition 2:

For every l, l >= ES, P(S ≥ l) is maximized when q1 = ··· = qn-mz = q = ES / (n – mz), and qn – mz + 1 = ··· = qn = 0, for some mz.

That is, in the terminology of Proposition 1, for all non-trivial cases, mo = 0, and thus the probability of deviation is maximized by some Bernoulli variable (rather than a shifted Bernoulli variable).
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