The boundedness–multiplicity-free module conjecture for reductive subalgebras

Let $\mathfrak{k}\subset\mathfrak{g}$ be a reductive in $\mathfrak{g}$ subalgebra. A $(\mathfrak{g},\mathfrak{k})$-module is multiplicity free if every simple $\mathfrak{k}$-module occurs in it with multiplicity at most one. Boundedness–multiplicity-free module conjecture. The subalgebra $\mathfrak{k}$ is bounded if and only if there exists a simple infinite-dimensional multiplicity-free $(\mathfrak{g},\mathfrak{k})$-module. The conjecture is presented as being supported by the empirical evidence collected in the paper; no resolution is given there.