Local-global principles for hermitian spaces over semi-global fields

Let $K$ be a complete discrete valued field with residue field $k$ and $F$ the function field of a curve over $K$. Let $A\in {}_{2}Br(F)$ be a central simple algebra with an involution $\sigma$ of any kind and $F_0=F^{\sigma }$. Let $h$ be an hermitian space over $(A,\sigma )$ and $G=SU(A,\sigma ,h)$ if $\sigma$ is of first kind and $G=U(A,\sigma ,h)$ if $\sigma$ is of second kind. Suppose that $\text{char}(k)\mathrm{\ne }2$ and ind$(A)\le 4$. Then we prove that projective homogeneous spaces under $G$ over $F_0$ satisfy a local-global principle for rational points with respect to discrete valuations of $F$.