Great find, Marc23! The bandeau pictured in the portrait of Princess Adelheid looks definitely like the bandeau worn by AD Regina and Francesca. On the RJWMB they speculate that this is the base of Empress Zita's tiara, but imho the resemblance is only vague. There are no pearls in Zita's tiara, eg., but there are in Adelheid's bandeau. And as Marengo already pointed out, the number of pearls, the proportions and the spacing are a match.
The portrait of Princess Adelheid looks like early twentieth century, when smaller bandeau tiaras had become en vogue. So it could very well be jewel that designed as a bandeau, and not just a base of a bigger tiara.