We describe a conjectural formula via intersection numbers for the
Masur-Veech volumes of strata of quadratic differentials with prescribed zero
orders, and we prove the formula for the case when the zero orders are odd. For
the principal strata of quadratic differentials with simple zeros, the formula
reduces to compute the top Segre class of the quadratic Hodge bundle, which can
be further simplified to certain linear Hodge integrals. An appendix proves
that the intersection of this class with ψ-classes can be computed by
Eynard-Orantin topological recursion.
As applications, we analyze numerical properties of Masur-Veech volumes, area
Siegel-Veech constants and sums of Lyapunov exponents of the principal strata
for fixed genus and varying number of zeros, which settles the corresponding
conjectures due to Grivaux-Hubert, Fougeron, and elaborated in [the7]. We also
describe conjectural formulas for area Siegel-Veech constants and sums of
Lyapunov exponents for arbitrary affine invariant submanifolds, and verify them
for the principal strata