I would say on balance it would firmly be in the not-proven camp. The rates of previous mass extinctions cannot be judged against modern events, for instance the date resolution of the Permian fossil record is greater than all human history.
The motivation of most of "The Sixth Extinction" hyperbole is firmly in the "We're all going to hell in handcart" school of environmentalism. The prediction record of that camp is frankly laughable, and the same often misanthropic tropes get dragged out each time.
There is another deeper question about why even should one care. Even if another extinction is in progress, so what? The timescales are immaterial to human civilisation, let alone an individual lifetime. Sure I love walking through a wildflower meadow and seeing birds and the beasts, but should my love for idea of untrammelled Amazon rainforest take precedence over a poor South American wanting a job or a US retiree whose pension fund is invested in palm oil futures? All things must pass, human civilisation will end, my nieces' and nephews' nieces and nephews will surely die. The world turns, the genes mutate and flow between their hosts and life goes on.
I often see it as a choice; to see the wonder and the beauty of the world and of human art, industry and inventiveness or to wallow in the venality of some and deplore the horror. It may all seem a bit Panglossian, but there is a rationality behind it all, historically the optimists are right more often than the pessimists.
