We study a strengthening of $MM^{++}$ which is called $MM^{*, ++}$ and which was introduced by Asperó and Schindler. We force its bounded version $MM^{*, ++}_{c}$, which is stronger that both $MM^{++}(c)$ as well as $BMM^{++}$, by $P_{max}$ forcing. We also give the construction of the ground model, which builds upon Gappo and Sargsyan, and the derived model construction of Larson, Sargsyan, and Wilson. This is a joint work with Ralf Schindler.