With the explosive development of big data, it is necessary to sort the data according to their importance or priorities. The sources with different importance levels can be modeled by the multilevel diversity coding systems (MDCS). Another trend in future communication networks, say 5G wireless networks and Internet of Things, is that users may obtain their data from all available sources, even from devices belonging to other users. Then, the privacy of data becomes a crucial issue. In a recent work by Li \etal, the secure asymmetric MDCS (S-AMDCS) with wiretap channels was investigated, where the wiretapped messages do not leak any information about the sources (\ie~ perfect secrecy). It was shown that superposition (source-separate coding) is not optimal for the general S-AMDCS and the exact full secure rate region was proved for a class of S-AMDCS. In addition, a bound on the key size of the secure rate region was provided as well. As a further step on the S-AMDCS problem, this paper mainly focuses on the key size characterization. Specifically, the constraints on the key size of superposition secure rate region are proved and a counterexample is found to show that the bound on the key size of the exact secure rate region provided by Li \etal~ is not tight. In contrast, tight necessary and sufficient constraints on the secrecy key size of the counterexample, which is the four-encoder S-AMDCS, are proved.