  Let $q$ be an anisotropic quadratic form defined over a general field $F$. In
this article, we formulate a new upper bound for the isotropy index of $q$
after scalar extension to the function field of an arbitrary quadric. On the
one hand, this bound offers a refinement of a celebrated bound established in
earlier work of Karpenko-Merkurjev and Totaro; on the other, it is a direct
generalization of Karpenko's theorem on the possible values of the first higher
isotropy index. We prove its validity in two important cases: (i) the case
where $\mathrm{char}(F) \neq 2$, and (ii) the case where $\mathrm{char}(F) = 2$
and $q$ is quasilinear (i.e., diagonalizable). The two cases are treated
separately using completely different approaches, the first being
algebraic-geometric, and the second being purely algebraic.
