Geometric lifts conjecture for zero-dimensional homology interleavings

Let $k$ be either $$\mathbb Q$$ or $$\mathbb Z/p\mathbb Z$$ for some \prime $p$. Let $M$ and $N$ be $$\epsilon$-interleaved$B_n$-modules, and suppose that there are topological spaces$X_M$and$X_N$ and functions

$$f_M:X_M\to \mathbb R^n,\qquad f_N:X_N\to \mathbb R^n$$

such that

$$H_0\circ F^S(X_M,f_M)\cong M,\qquad H_0\circ F^S(X_N,f_N)\cong N.$$

Geometric lifts conjecture. There exists a CW-complex $X$ and continuous functions $$\gamma_M, $$ $\gamma_N:X\to \mathbb R^n$ $ such that

$$H_0\circ F^S(X,\gamma_M)\cong M,\qquad H_0\circ F^S(X,\gamma_N)\cong N,$$

and

$$\|\gamma_M-\gamma_N\|_{\infty}=\epsilon.$$

This conjectural extension of the geometric-lift proposition from positive homological degree to $i=0$ would imply the corresponding extension of the optimality theorem for the interleaving distance. The source presents it as conjectural, and no resolution is supplied here.