Why is it recommended (Mandub) in the Hanafi school to perform Wudu after touching a woman's skin without a barrier?

Because it breaks Wudu in the Hanafi school.
To escape scholarly disagreement (al-Khuruj min Khilaf), as it breaks Wudu in the Shafi'i school.
Because women's skin is physically impure.
Because the Prophet ﷺ never did it.