Conjecture on maximal cohomology for generalized 12-fold tilings

Let $\Omega_{E_{12}^{\gamma}}$ be the tiling space associated with a generalized 12-fold tiling with parameter $\gamma$. For generic parameters, meaning those with the maximal number of singular lines and the least order of multiple intersections of singular lines, the paper considers the cohomology groups $H^i(\Omega_{E_{12}^{\gamma}})$. Maximal cohomology conjecture. The maximal cohomology attained among all the generalized 12-fold tilings is

$$H^0(\Omega_{E_{12}^{\gamma}})=\mathbb Z,\quad H^1(\Omega_{E_{12}^{\gamma}})=\mathbb Z^{25}\quad\mathrm{and\ }\quad H^2(\Omega_{E_{12}^{\gamma}})=\mathbb Z^{564}.$$

The theorem preceding this conjecture establishes that the cohomology groups are torsion free and that the rank of $H^1(\Omega_{E_{12}^{\gamma}})$ can attain $25$; the maximality assertion for the cohomology, particularly the stated value of the second cohomology rank, is proposed on the basis of computer calculations. Special parameter values can cause multiple intersections of singular lines to merge, so determining all possible second-cohomology ranks remains computationally involved.