Balanced-refinement core conjecture for Lipschitz selections

Let $({\mathcal M},\rho)$ be a pseudometric space, let $X$ be a Banach space, and let $m$ be a fixed positive integer. Set

$$\ell=\min\{m+1,\dim X\},\qquad N(m,X)=2^{\ell}.$$

For a set-valued mapping $F:{\mathcal M}\to{\mathcal K}_m(X)$, define its iterated balanced refinements by

$$F^{[0]}=F,\qquad F^{[k+1]}(x)=\bigcap_{z\in{\mathcal M}}\left[F^{[k]}(z)+\lambda_{k+1}\rho(x,z)B_X\right].$$

Here a set-valued mapping $G:{\mathcal M}\to{\mathcal K}_m(X)$ is a $\gamma$-core of $F$ when $G(x)\subset F(x)$ and $\operatorname{d_H}(G(x),G(y))\le\gamma\rho(x,y)$ for all $x,y\in{\mathcal M}$. Balanced-refinement core conjecture. There exist $\gamma\ge1$ and numbers $\lambda_1,\ldots,\lambda_{\ell}\ge1$ such that, whenever every restriction of $F$ to a subset of ${\mathcal M}$ of cardinality at most $N(m,X)$ has a Lipschitz selection with seminorm at most $1$, the mapping $F^{[\ell]}$ is a $\gamma$-core of $F$. The conjecture proposes a finite iterative construction of a Lipschitz-controlled core from the local finiteness hypothesis underlying the Lipschitz selection principle.