The pre-Tango structure is a certain invertible sheaf of locally exact differentials on a curve in positive characteristic. On any curve of sufficiently high genus, there necessarily exist pre-Tango structures. Meanwhile, by using the notion of pre-Tango structure, we can construct a form of the affine line over the curve. The completions of the forms are regarded as a generalization of Raynaud's counter-example to the Kodaira vanishing theorem. This suggests that we may have certain pathological phenomena on the completions of all such forms.
For the time being, we consider whether every curve of genus greater than one has a pre-Tango structure which brings certain pathological phenomena. In the present article, we give a sufficient condition for the completion of the form which is induced from a pre-Tango structure to have non-closed global differential 1-forms. Moreover, we give a lower bound for the dimension of the locus of the curves which have pre-Tango structures satisfying that sufficient condition, in the moduli space of curves.