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We investigate connections between the sGG property of compact complexmanifolds, defined in earlier work by the second author and L. Ugarte by therequirement that every Gauduchon metric be strongly Gauduchon, and a possibledegeneration of the Fr\"olicher spectral sequence. In the first approach thatwe propose, we prove a partial degeneration at $E_2$ and we introduce apositivity cone in the $E_2$-cohomology of bidegree $(n-2,\,n)$ of the manifoldthat we then prove to behave lower semicontinuously under deformations of thecomplex structure. In the second approach that we propose, we introduce ananalogue of the $\partial\bar\partial$-lemma property of compact complexmanifolds for any real non-zero constant $h$ using the partial twisting $d_h$,introduced recently by the second author, of the standard Poincar\'edifferential $d$. We then show, among other things, that this$h$-$\partial\bar\partial$-property is deformation open.

Positivity Cones under Deformations of Complex Structures