  We show that on a compact K\"ahler manifold all real $(1,1)$-classes
admitting solutions to the supercritical deformed Hermitian-Yang-Mills equation
form a both open and closed subset of those which satisfy the numerical
condition proposed by Collins-Jacob-Yau. More importantly, we show by examples
that it can be a proper subset. This disproves a conjecture made by
Collins-Jacob-Yau.
