BB Fan 72

R2, less than 2 outs. Batter hits deep to centerfold, R2 with a large lead, takes off on the hit, touches third and takes 3-4 steps towards home before seeing that the center fielder has a play on the ball. R2 returns towards second. (A) does not retouch 3rd or (B) retouches third. As the R2 gets half-way back to second, the center fielder drops the ball. R2 then runs directly home (crossing the edge of the mound). The defense then makes a proper appeal for not touching third. Is R2 out for either case A or B?