  We associate to any complete spherical variety $X$ a certain nonnegative
rational number $\wp(X)$, which we conjecture to satisfy the inequality $\wp(X)
\le \operatorname{dim} X - \operatorname{rank} X$ with equality holding if and
only if $X$ is isomorphic to a toric variety. We show that, for spherical
varieties, our conjecture implies the generalized Mukai conjecture on the
pseudo-index of smooth Fano varieties due to Bonavero, Casagrande, Debarre, and
Druel. We also deduce from our conjecture a smoothness criterion for spherical
varieties. It follows from the work of Pasquier that our conjecture holds for
horospherical varieties. We are able to prove our conjecture for symmetric
varieties.
