On property $D(2)$ and common splitting field of two biquaternion algebras

Let $F$ be a field of characteristic $\ne 2$. We say that $F$ has property $D(2)$ if for any quadratic extension $L/F$ and any two binary quadratic forms over $F$ having a common nonzero value over $L$ this value can be chosen in $F$. There exist examples of fields of characteristic 0 which do not satisfy property $D(2)$. However, as far as we know, such examples of positive characteristic have not been constructed.
In this note we show that if $k$ is a field of characteristic $\ne 2$ such that $\|k^*/{k^*}^2\|\ge 4$, then for the field $k(x)$ property $D(2)$ does not hold. Using this we construct two biquaternion algebras over a field $K=k(x)((t))((u))$ such that their sum is a quaternion algebra, but they do not have a common biquadratic (i.e. a field of the kind $K(\sqrt a,\sqrt b)$, where $a,b\in K^*$) splitting field.