Are you saying the pipe lifts off that first support on the horizontal run in the operating state?
While liftoff may be acceptable, it is not the best support situation. The deadweight loads shifts to other supports (and the nozzle). And the system may be less stable in the horizontal direction.
Reread your text. Does it say any nozzle that moves 13 mm requires a spring hanger or does it say if support is necessary, then a spring should be used. My guess is the latter.
My opinion is, if you check stresses and loads and deflections and all are acceptable in the installed and operating conditions without the spring and if there is no stability issue, then no spring is required.
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Dave Diehl