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We show that if A is a simple Villadsen algebra of either the first type with seed space a finite dimensional CW complex, or of the second type, then $A$ absorbs the Jiang-Su algebra tensorially if and only if the central sequence algebra of A does not admit characters. Additionally, in a joint appendix with Joan Bosa, we show that the Villadsen algebra of the second type with infinite stable rank fails the Corona Factorization Property, thus providing the first example of a unital, simple, separable and nuclear $C^\ast$-algebra with a unique tracial state which fails to have this property.

Regularity of Villadsen algebras and characters on their central sequence algebras