Universality of the interleaving distance over arbitrary fields

Let $k$ be a field, $n \ge 1$ and $i \ge 0$. An $n$-parameter persistence module is a functor $M\colon (\mathbb{R}^n, \le) \to \mathrm{Vect}_k$, for the coordinatewise order. Write $\mathbf{1} = (1, \dots, 1)$, $M(\varepsilon)_a = M_{a + \varepsilon \mathbf{1}}$, and let $\eta_M^{2\varepsilon}\colon M \to M(2\varepsilon)$ be the structure morphism. Modules $M$ and $N$ are $\varepsilon$-interleaved if there are natural transformations $\varphi\colon M \to N(\varepsilon)$ and $\psi\colon N \to M(\varepsilon)$ with $$\psi(\varepsilon) \circ \varphi = \eta_M^{2\varepsilon}, \qquad \varphi(\varepsilon) \circ \psi = \eta_N^{2\varepsilon},$$ and the interleaving distance $d_I(M,N)$ is the infimum of such $\varepsilon \ge 0$, or $+\infty$ if there is none. For a topological space $X$ and a function $f\colon X \to \mathbb{R}^n$, not necessarily continuous, let $P_i(f)$ be the module $a \mapsto H_i(\{x \in X : f(x) \le a\}; k)$ of singular homology, with the maps induced by the inclusions of sublevel sets, and let $\mathcal{R}_{n,i}(k)$ be the class of modules isomorphic to some $P_i(f)$. An extended pseudometric $d$ on isomorphism classes of $n$-parameter modules (which may take the value $+\infty$ and vanish on distinct classes) is $i$-stable if $$d(P_i(f), P_i(g)) \le \sup_{x \in X} \|f(x) - g(x)\|_\infty$$ for every space $X$ and all $f, g\colon X \to \mathbb{R}^n$, the supremum over an empty $X$ being $0$. Lesnick proved that $d_I$ is $i$-stable for every $i$, and that every $i$-stable extended metric (one that is positive on distinct classes) is bounded by $d_I$ on $\mathcal{R}_{n,i}(k)$ when $k$ is a prime field and $i \ge 1$.

Problem. Is it true that for every field $k$, every $n \ge 1$ and every $i \ge 0$, every $i$-stable extended pseudometric $d$ satisfies $$d(M,N) \le d_I(M,N) \qquad \text{for all } M, N \in \mathcal{R}_{n,i}(k)?$$ No finite-dimensionality of the modules is assumed.