On Shimura curves in the Torelli locus of curves

Oort has conjectured there do not exist Shimura curves lie generically of Torelli locus of curves of genus $g \geq 8$.
We show that  there do not exist Shimura families of semi-stable curves of genus $g\geq 5$ over curves of Mumford type.

We also show that there do not exist Shimura curves, which lie generically in the Torelli locus of hyperelliptic curves of genus $g\geq 8$.
The first result proves a slight weaker form of the conjecture for the case of Shimura curves of Mumford type.
The second result proves the conjecture for the Torelli locus of hyperelliptic curves.