Is there a skirmish underway in our skies among two Anunnaki factions as claimed by the Pleiadians?

This in fact is true. It is happening in outposts where there is more sparse presence of craft that keeps the relative balance of power more even than would be the case of a challenge on or near the home planet Nibiru. The other members of the Extraterrestrial Alliance give this a wide berth because they wish to have no combat interaction with the Anunnaki belonging to either faction and, if anything, would side with the established order under Anu because that is the lineage they have an alliance with—the others are interlopers. But the extraterrestrials across the board, being sociopaths, are opportunistic and, in the end, will abandon ship if they conclude it serves them. But they are clever enough to know that there could be a huge Anunnaki reprisal for siding with the enemy and do not wish to take part in war against the Anunnaki in full force.