Minimal persistence grids versus homological Morse lower bounds

Fix a field $\Bbbk$, an integer $n \ge 1$ and a finite nonempty simplicial complex $K$ of dimension $d$. A filter assigns to every nonempty simplex a vector $f(\sigma) \in [0,1]^n$, with $f(\tau) \le f(\sigma)$ coordinatewise whenever $\tau \subseteq \sigma$. Put $$K_u = \{\sigma \in K : f(\sigma) \le u\}, \qquad M^q_u = H_q(K_u; \Bbbk), \qquad \mathbf M = (M^0, \ldots, M^d),$$ with ordinary unreduced homology and the maps induced by inclusions. Let $\xi_i^q(u)$ be the multigraded Betti numbers of the persistence module $M^q$: the number of free summands generated at grade $u$ in homological position $i$ of a minimal free resolution, computed after restricting to a finite grid determining the module and regrading its coordinate sets into $\mathbb{Z}^n$; set $\xi_i^q = 0$ for $q < 0$ or $q > d$. The minimal grid of $\mathbf M$ is $G = G_1 \times \cdots \times G_n$, where $$G_j = \bigcup_{q=0}^{d} \{u_j : \xi_0^q(u) + \xi_1^q(u) > 0\},$$ so that the values $0$ and $1$ belong to $G_j$ only if they occur there. Put $$b_q(u) = \max\Bigl\{0,\ \xi_0^q(u) + \xi_1^{q-1}(u) - \sum_{p=1}^{n-1} \xi_{p+1}^{q-p}(u)\Bigr\}, \qquad \mathcal L(\mathbf M) = \sum_{u \in G} \sum_{q=0}^{d} b_q(u).$$

Harrington, Tillmann and Torras-Pérez bound the dimension of the space of filters with persistent homology $\mathbf M$ by $n|K| - \sum_j |G_j|$ and by $\frac n2 (|K| - \mathcal L(\mathbf M))$, where $|K|$ is the number of simplices; the second bound is at least as strong as the first whenever $|G_j| \le \mathcal L(\mathbf M)$ for every $j$. They note that this holds for $n = 1$ and for filters injective in every coordinate, for which $\mathcal L(\mathbf M) = |K|$.

Problem. Does every such filter satisfy $$|G_j| \le \mathcal L(\mathbf M) \qquad \text{for every } j = 1, \ldots, n?$$ The tuple $\mathbf M$ is the persistent homology of a filter on a finite simplicial complex, not an arbitrary tuple of persistence modules, and the filter may assign equal coordinate values to distinct simplices.