From the original post:

"The concept of private property is, and always has been, hotly debated among anarchists. Many believe resources should be available to all, while others prefer they be in the hands of the most efficient. Anarcho-capitalists assert that those who most efficiently supply other people’s needs would and should command a higher market share relative to less efficient market participants. Market anarchists agree that those who use resources most efficiently should be allowed to use their justly obtained property to accumulate more, if that’s what they so desire."

The line "Market anarchists agree that those who use resources most efficiently should be allowed to use their justly obtained property to accumulate more..." seems to suggest that Market Anarchist have some prerequisite for private property, and that there is an "efficient" class that has this privilege.