Stick principle with failure of SCH

Establish whether there exists a model in which \(\lambda\) is supercompact, a normal ultrafilter \(\mathscr U\) on \(\lambda\) satisfies \({\rm Gal}(\mathscr U,\lambda^+,\lambda^+)\), the stick principle \((\lambda)\) holds, and \(2^\lambda>\lambda^+\).

Background

The paper’s stick-based approach proves that $(\lambda)$ implies the negative relation $\lambda^+\nrightarrow(\lambda^+,(3)_{\lambda})^2$. The authors discuss forcing over a supercompact cardinal using a normal ultrafilter with the Galvin property to preserve old subsets of size $\lambda^+$ through Prikry-type forcing.

The unresolved issue is whether the required ultrafilter property, stick, and failure of SCH can be obtained simultaneously. Such a model would provide a forcing-theoretic realization of the stick approach in the cardinal-arithmetic regime relevant to the paper.