Eleazar ben Annas = Alexander of Acts 4.6?

I recently read that Alexander is the Graecized form of the Hebrew name Eleazar. We know that Annas the high priest of 6-15AD had five sons who served as high priest, one of whom was named Eleazar. Since we have no record of a high priest bearing the name Alexander in Josephus, et al., and since we are told that one Alexander was of "the high priestly family" in Acts 4.6, might that Alexander mean to be a reference to Eleazar ben Annas, who served as high priest from 16-17AD?

Additionally, the John of Acts 4.6 might very well be Jonathan ben Annas, who served as high priest twice. This would explain further why Luke can call this group members of "the high priestly family". If Luke's recipient Theophilus is the son of Annas (as is posited here on this blog) who served as high priest from 37-41AD, he would have easily recognized his brothers' names, even if shortened or Graecized. Perhaps there is an intention on Luke's part to publicly obscure, yet reveal to Theophlius, the identities of these "high priests". I cannot find any reason for the phenomenon known as "protective anonimity" in this case. However, the irony is far more than coincidence.

Also, note that Luke does not say that these men are of "priestly descent". He says they are of the "HIGH priestly family". This is a much more specific reference. For there is no basis of descent which entitles one to the high priesthood.