Recently Moti Gitik refuted Shelah's PCF conjecture, by producing a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1.$ See his papers Short extenders forcings I and Short extenders forcings II.

In Gitik's model the cardinal $\kappa=\sup(a)$ is a fixed point of the $\aleph$-function.

Question. Can we improve Gitik's result to get a countable set $a$ of regular cardinals with $|\operatorname{pcf}(a)| \geq \aleph_1$ such that $\kappa=\sup(a)$ is not a fixed point of the $\aleph$-function?