Thomas's multirank conjecture for linearizations

Let $\mathscr{P}$ be a finite poset, and let $B_1$ and $B_2$ be $\mathscr{P}$-persistence sets. Write $L_{B_1}$ and $L_{B_2}$ for their linearizations, and let $\mathrm{multirank}_{L_B}(S_1,S_2)$ denote the multirank associated with a pair of slices $(S_1,S_2)$. Thomas's multirank conjecture. If

$$\mathrm{multirank}_{L_{B_1}}(S_1,S_2)=\mathrm{multirank}_{L_{B_2}}(S_1,S_2)$$

for all pairs of slices $(S_1,S_2)$, then

$$L_{B_1}\cong L_{B_2}.$$

The statement asserts that multirank invariants completely determine the linearizations of finite-poset persistence sets. In the paper it is presented as Thomas's conjecture and is proved for the setting under consideration, so the claim is solved.