Let $d,q \in \mathbb{N}$, and let $f:\mathrm{GF}(q) \to \mathrm{GF}(q)$ be a univariate polynomial. In this case, it is possible to test whether $f$ is of degree at most $d$ (or whether $f$ is at least $\epsilon$ from it) using the algorithm of (RS96). The testing algorithm works by simply trying to interpolate the function $f$ on $\Theta(1/\epsilon)$ collections of $d + 2$ uniformly selected points, and checking whether the resulting functions are all polynomial of degree at most $d$.
Is there an extension of this result that allows to test whether a bivariate polynomial $g:\mathrm{GF}(q)^2 \to \mathrm{GF}(q)$ is of degree at most $d$ in the first variable, using $O(\epsilon^{-1} \cdot d)$ queries?