  We are interested in stringy invariants of singular projective algebraic
varieties satisfying a strict monotonicity with respect to elementary
birational modifications in the Mori program. We conjecture that the algebraic
stringy Euler number is one of such invariants. In the present paper, we prove
this conjecture for varieties having an action of a connected algebraic group G
and admitting equivariant desingularizations with only finitely many G-orbits.
In particular, we prove our conjecture for arbitrary projective spherical
varieties.
