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We study the spectrum in such a PT-symmetric square well of a diameter Lwhere the "strength of the non-Hermiticity" is controlled by the twoparameters, viz., by an imaginary coupling ig and by the distance d of itsonset from the origin. We solve this problem and confirm that the spectrum isdiscrete and real in a non-empty interval of g. Surprisingly, a specificdistinction between the bound states is found in their asymptoticstability/instability with respect to an unlimited growth of g. In our model,all of the low-lying levels remain asymptotically unstable at the small d andfinite L while only the stable levels survive for d near L or in the purelyimaginary well with infinite L. In between these two extremes, an unusual andtunable, variable pattern of the interspersed "robust" and "fragile" subspectraof the real levels is obtained.Comment: final version: 33 pages (plus the old 8 figures of version 1

Solvable PT-symmetric model with a tunable interspersion of nonmerging levels