Does this behavior still have something to do with the warrants?

I don't think that shares are being sold to move into the warrants, not on a large scale, anyway - Share volume is much larger than warrant volume. At these prices, there is more leverage in the warrants that the shares. If/when the share price hits $5, the warrants will be worth over $4, so there is approximately twice the leverage in the warrants to the common stock.

( $4/ $0.28 = 14.3 vs $5/ $0.60 = 8.3 ).

Cheers