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](http://www.math.tau.ac.il/~gitik/se1.pdf) and  [Short extenders forcings II](http://www.math.tau.ac.il/~gitik/short%20extenders%20forcing%202-2015.pdf).

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?