  Can a supercompact cardinal kappa be Laver indestructible when there is a
level-by-level agreement between strong compactness and supercompactness? In
this article, we show that if there is a sufficiently large cardinal above
kappa, then no, it cannot. Conversely, if one weakens the requirement either by
demanding less indestructibility, such as requiring only indestructibility by
stratified posets, or less level-by-level agreement, such as requiring it only
on measure one sets, then yes, it can.
