This is a question that I've posted to Mathematics Stack Exchange, that was un-answered.

To re-iterate it here:

Is the following known to be consistent relative to some large cardinal assumption?

$\forall \kappa [\kappa >2 \to \kappa < \kappa^* < 2^\kappa \wedge singular(\kappa^*)]$

where $\kappa$ is a cardinal and $``<"$ is strict cardinal smaller than, and $\kappa^* = |\{\lambda|\kappa < \lambda < 2^\kappa\}|$