Concentration of measure on finite powers of $S^\infty$

Question

I am wondering about a natural generalization of theorem 1.4 in the article Dvoretzky's theorem — Thirty years later by Milman. My first thought was to look at Milman's paper that he cites for the result (On a property of functions defined on infinite-dimensional manifolds (MSN)), but I have been unable to track it down.

Here is the generalization I'm wondering about:

Let $S^\infty$ be the unit sphere of $\ell_2$ with real coefficients. Fix a finite integer $k$. Is it true that for any uniformly continuous function $f: (S^\infty)^k \rightarrow \mathbb{R}$ there exists a continuous function $\varphi : [-1,1]^{k \choose 2} \rightarrow \mathbb{R}$ such that for any $\varepsilon >0$ and $n$ there is an $n$-dimensional subspace $E$ of $\ell_2$ such that for any $x_0,\dots,x_{k-1}$ in the unit sphere of $E$, $$|f(x_0,x_1,\dots,x_{k-1})-\varphi\left(r_{0,1},r_{0,2},\dots,r_{0,k-1},r_{1,2},\dots,r_{k-2,k-1}\right)| < \varepsilon,$$ where $r_{i,j}=\left<x_i,x_j \right>$ for any $i<j<k$?

Basically, to any given degree of accuracy $\varepsilon$ are there arbitrarily large linear subspaces on which $f(x_0,\dots,x_{k-1})$ only depends on the inner products of the $x_i$ to within $\varepsilon$?

Answer 1

I was able to track down the paper and after Theorem 3, which is the result in question in the $k=2$ case he says:

Аналогичный теореме 3 факт имеет место и для функции к переменных. Приведем формулировку для трех переменных, поскольку дальнейшие обобщения очевидны.

Which a certain website translates as:

A fact similar to Theorem 3 holds for the function $k$ of variables. We give the statement for three variables, since further generalizations are obvious.

Which (modulo the misplaced $k$) seems to be saying that Milman does claim the generalization I'm after.