Stick, a Galvin ultrafilter, and failure of SCH

Determine whether it is consistent that a cardinal \(\lambda\) is supercompact, a normal ultrafilter \(\mathscr U\) on \(\lambda\) satisfies \({\rm Gal}(\mathscr U,\lambda^+,\lambda^+)\), the stick principle \(\clubsuit(\lambda)\) holds, and \(2^\lambda>\lambda^+\).

Background

The paper shows that the stick principle $\clubsuit(\lambda)$ implies the desired negative partition relation. It discusses forcing $\clubsuit(\lambda)$ while increasing $2^\lambda$, and separately describes a Galvin-type property for a normal ultrafilter that can preserve stick or tiltan through Prikry-type forcing.

The unresolved issue is whether these ingredients can be forced together: supercompactness, the Galvin ultrafilter property, stick at $\lambda^+$, and failure of SCH at $\lambda$.