Moduli spaces and conic bundles

Question

The moduli space $A_2(1,8)^{\operatorname{lev}}$ of $(1,8)$-polarized abelian surfaces with canonical level structure has a structure of conic bundle over $\mathbb{P}^2$ with a curve of degree $4$ as discriminant. In, particular $A_2(1,8)^{\operatorname{lev}}$ is rational. See Section $6$ in

M. Gross and S. Popescu, Calabi–Yau threefolds and moduli of abelian surfaces. I, Compositio Mathematica, 127 (2001), no. 2, 169–228.

I wanted to ask if there are examples of non rational $3$-fold moduli spaces, not necessarily parametrizing abelian surfaces, that are birational to a conic bundle over a rational surface?

Thank you very much.

Answer 1

here

https://arxiv.org/pdf/1409.5033.pdf

in section 5 there's an example of unirational, non-rational, 3fold moduli space. I am not sure whether it is a conic bundle, though. Probably not.