Context
The Kirszbraun theorem states that if $H_1$ and $H_2$ are Hilbert spaces, and $E \subset H_1$, then any Lipschitz function $f: E \to H_2$ may be extended to a Lipschitz function $\widehat{f}: H_1 \to H_2$ with the same Lipschitz constant.
When the codomain is $\mathbb{R}$, this is done by defining, $$\widehat{f}(x) := \inf_{y \in E} \big(f(y) + \operatorname{Lip}(f) \cdot |x - y | \big).$$ This is remarkably similar to the manner in which E. J. McShane extends a uniformly continuous function on a subset of a metric space $E \subset M$, $g: E \to \mathbb{R}$ with concave modulus of continuity $\omega$. Furthermore, his extension has the same modulus of continuity and is defined similarly to the Lipschitz case: $$\widehat{g}(x) := \sup_{y \in E} \big(g(y) - \omega(d(x, y))\big).$$ (I'm matching the two definitions of the sources, but one can pick either the $\inf$ or $\sup$ for either definition with the appropriate $\pm$).
My Question
My question is whether it is known (and in what direction) if one may extend uniformly continuous functions on Hilbert Spaces maintaining the modulus of continuity (not just when the codomain is $\mathbb{R}$). If it's impossible, is there an easy counter-example (say, in $\mathbb{R}^2$)?
Some Thoughts
If one is in finite dimensions (the case I actually care about), then you can certainly extend using the result with codomain $\mathbb{R}$ on each coordinate map. However, (I think) this will leave your new extension with an extra factor of $\sqrt{d}$. I'm hoping for a claim that maintains the modulus of continuity exactly.
To be clear, I'm honestly not asking for a proof or anything. Mostly it's that this sort of theory is outside of my expertise, so I don't know which references to even look around in. Advice on that front would be sufficient for me.
This question is related to this StackExchange post.
