There is no bound on the order of the conditions needed to write a multisymplectic form with constant coefficients: for every j, explicit 3-forms in dimension 2j+5 require conditions of order j.
On the Geometry of Multisymplectic Manifolds
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The Spencer cohomology and integrability of multisymplectic structures
There is no bound on the order of the conditions needed to write a multisymplectic form with constant coefficients: for every j, explicit 3-forms in dimension 2j+5 require conditions of order j.