  The purpose of this work is to evidence a pathological set of initial data
for which the regularized solutions by convolution experience a norm-inflation
mechanism, in arbitrarily short time. The result is in the spirit of the
construction from Sun and Tzvetkov, where the pathological set contains
superposition of profiles that concentrate at different points. Thanks to
finite propagation speed of the wave equation, and given a certain time, at
most one profile exhibits significant growth. However, for Schr\"odinger-type
equations, we cannot preclude the profiles from interacting between each other.
Instead, we propose a method that exploits the regularizing effect of the
approximate identity which, at a given scale, rules out the norm inflation of
the profiles that are concentrated at smaller scales.
