The linearization maximum conjecture for set-system dimension

A set system is a family of sets, and its dimension $\dim \mathcal{X}$ is its learning-sequence order type. A linearization of a set system is a suitable set system preserving the relevant union structure and linearly ordered by inclusion. The paper asks whether

$$\dim \mathcal{X}=\max\left\{\dim \mathcal{Y}\;;\;\mathcal{Y}\ \text{is a linearization of }\mathcal{X}\right\}$$

for every set system $\mathcal{X}$. Linearization maximum conjecture. The displayed equality holds for every set system $\mathcal{X}$. This seeks a set-system analogue of the de Jongh–Parikh theorem for well-quasi-orders; the source formulates it as a question and does not provide a resolution.