Shelah's diamond principle conjecture for stationary subsets

Let $\lambda$ and $\kappa$ be cardinals, with $\lambda=\lambda^{<\lambda}>\kappa^+$, $\kappa=\operatorname{cf}(\kappa)$, and $\lambda\ne\aleph_1$. The weaker alternative allows $\lambda\ge\beth_\omega$ instead of $\lambda\ne\aleph_1$. Write $(D\ell)_{S^\lambda_\kappa}$ for the corresponding diamond-like principle on $S^\lambda_\kappa$. The conjecture. Under these cardinal-arithmetic assumptions, $(D\ell)_{S^\lambda_\kappa}$ holds. This positive answer would imply $\operatorname{TDU}_{\aleph_{\omega+1}}$; the source presents the assertion as an open question in its discussion of consequences of weak generalized continuum-hypothesis assumptions.