A theorem by Solovay says that if $\kappa$ is strongly compact then the singular cardinal hypothesis holds above $\kappa$. But if we only assume that $\kappa$ is $\lambda$-compact (edit: or $\lambda^+$-compact) for some singular $\lambda$, does it follow that the singular cardinal hypothesis holds at $\lambda$?
Edit: My motivation is that I wonder if the $\lambda$-supercompactness of $\kappa$, if $\lambda$ is a strong limit cardinal and $cof (\lambda) \lt \kappa$, implies that $\kappa$ is $2^{\lambda}$-supercompact, as I've read somewhere that $\lambda$-supercompact implies $\lambda^+$-supercompact if $cof (\lambda) \lt \kappa$.
The paper "Strong axioms of infinity and elementary embeddings" by Solovay, Reinhardt and Kanamori proves this result for supercompact cardinals in the discussion after theorem 3.4 (it says that if $\kappa$ is supercompact, $\lambda$ is regular and $\lambda \ge \kappa$, then $\lambda = \lambda^{\lt \kappa}$, but if we set $\lambda = \mu^+$ such that $cf (\mu) \lt \kappa$, then we get $\mu^{\lt \kappa} = \mu^+$, that is the singular cardinal hypothesis at $\mu$), and it's clear that the proof only needs $\kappa$ to be $\mu^+$-supercompact.
The paper also proves (proposition 3.2) that if $\kappa$ is $\lambda$-supercompact, it is $\lambda^{\lt \kappa}$-supercompact. I looked up that paper because a Mathoverflow answer cited it as source for the latter result.