Cubic bound for rank-exact Betti numbers of two-parameter persistence modules

Let $M: \mathbb{R}^2 \xrightarrow{\;\;\;} \operatorname{vec}$ be a finitely presented two-parameter persistence module. Write $\mathsf{b}(M)$ for the size of its usual Betti numbers and $\mathsf{b}^{\mathrm{rk}}(M)$ for the size of its Betti numbers relative to the rank exact structure.

Cubic Betti-number bound. We have

$$\mathsf{b}^{\mathrm{rk}}(M) \in O\left(\mathsf{b}(M)^3\right),$$

in the sense that $\mathsf{b}^{\mathrm{rk}}(M)$ can be bounded above by $\mathsf{b}(M)^3$ times a constant independent of $M$.

The preceding examples show that the rank-exact Betti-number size can be substantially smaller than the usual Betti-number size, while the conjecture proposes a uniform cubic upper bound in the two-parameter case.