We prove that-consistently-in the space ω* there are no P-sets with the ℂ-cc and any two fat P-sets with the ℂ⁺-cc are coabsolute.

Consider the poset $P_I=\text{Borel}\left(ℝ\right)\setminus I$ where $I$ is an arbitrary $\sigma$-ideal $\sigma$-generated by a projective collection of closed sets. Then the $P_I$ extension is given by a single real $r$ of an almost minimal degree: every real $s\in V\left[r\right]$ is Cohen-generic over $V$ or $V\left[s\right]=V\left[r\right]$.