In [Rezk - Compactly generated spaces](https://ncatlab.org/nlab/files/Rezk_CompactlyGeneratedSpaces.pdf) a k-Hausdorff property is defined, between [weakly Hausdorff](https://topology.pi-base.org/properties/P000143) and [unique sequential limits](https://topology.pi-base.org/properties/P000099).

On the other hand, a stronger notion of k-Hausdorff between $T_2$ and [compacts are closed](https://topology.pi-base.org/properties/P000100) was already in use in the literature, e.g. [Lawson and Madison - Quotients of k-semigroups](https://doi.org/10.1007/BF02194829).

Has the weaker k-Hausdorff property appeared in the literature beyond Rezk's nlab note? The preamble seemed to indicate it was novel. See also some discussion at [Math.SE](https://math.stackexchange.com/questions/4760309/how-are-k-hausdorff-and-weakly-hausdorff-distinct "How are k-Hausdorff and weakly Hausdorff distinct?").