Convex $C^1$ extensions of $1$-jets from compact subsets of Hilbert spaces

Let $X$ denote a Hilbert space. Given a compact subset $K$ of $X$ and two continuous functions $f:K\to \mathbb{R}$, $G:K\to X$, we show that a necessary and sufficient condition for the existence of a convex function $F\in C^1(X)$ such that $F=f$ on $K$ and $\mathrm{\nabla }F=G$ on $K$ is that the $1$-jet $(f,G)$ satisfies (1) $f(x)\ge f(y)+⟨G(y),x-y⟩$ for all $x,y\in K$, and (2) if $x,y\in K$ and $f(x)=f(y)+⟨G(y),x-y⟩$ then $G(x)=G(y)$. We also solve a similar problem for $K$ replaced with an arbitrary bounded subset of $X$, and for $C^1(X)$ replaced with the class $C_b^{1,u}(X)$ of differentiable functions with uniformly continuous derivatives on bounded subsets of $X$.