Failure of the local-global principle for quadratic forms over function fields

Let $K$ be a finitely generated field of transcendence degree $n\geq 2$ over an algebraically closed field $k$ of characteristic $\neq 2$. A quadratic form is isotropic if it admits a nontrivial zero. For a discrete valuation $v$ on $K$, let $K_v$ be the fraction field of the completion of its valuation ring. The local-global principle for isotropy in dimension $d$ over $K$ asserts that every quadratic form in $d$ variables over $K$ that is isotropic over $K_v$ for every discrete valuation $v$ on $K$ is isotropic over $K$.

Failure of the local-global principle. The local-global principle for isotropy of quadratic forms fails to hold in dimension $2^n$ over $K$.

This extends the known failure in transcendence degree two and gives the stated obstruction in every transcendence degree $n\geq 2$ over an algebraically closed field of characteristic different from $2$.