The maximality conjecture for $\Gamma_{\aleph_3}$

Let $\Gamma_{\aleph_3}$ be a class of partial orders, and let maximality with respect to a theory mean the five conditions given in the preceding definition: inclusion of the relevant forcing class, consistency preservation, inclusion of the canonical forcing notions, closure under locally complete embeddings, and maximality among definable classes. Maximality conjecture for $\Gamma_{\aleph_3}$. There is a class $\Gamma_{\aleph_3}$ which is maximal with respect to the theory

$${\text{{\sf ZFC}}}+{\text{{\sf MM}}}^{++}+\emph{ large cardinals}.$$

The conjecture proposes a maximal forcing class at $\aleph_3$ compatible with ${\text{{\sf MM}}}^{++}$ and large cardinals. The supplied text does not state any resolution or partial result for this assertion.