Hausdorff spaces with lattice isomorphism between the topologies [closed]

Question

For $i=1,2$ let $(X_i,\tau_i)$ be Hausdorff spaces such that the lattices $\tau_1, \tau_2$ are isomorphic.

Does this imply that $(X_1, \tau_1) \cong (X_2,\tau_2)$?

(This is a follow-up question to $T_2$-spaces such that the lattices of open sets can be embedded into each other.)