Tight $σ$-filtrations and the BFPP of $C(ω^*)$

Assuming that $\mathcal{P}(ω)$ is tightly $σ$-filtered, we prove that $C(ω^*)$ fails the ball fixed point property. This axiom holds under the continuum hypothesis and in the $\aleph_2$-Cohen model. The proof shows that the Stone space of the countable--cocountable algebra $\operatorname{Cc}(ω_1)$ i