Hyperbolicity in non-metric cubical small-cancellation

Given a non-positively curved cube complex $X$, we prove that the quotient of ${\pi }_{1}X$ defined by a cubical presentation $⟨X\mid Y_1,\dots ,Y_s⟩$ satisfying sufficient non-metric cubical small-cancellation conditions is hyperbolic provided that ${\pi }_{1}X$ is hyperbolic. This generalises the fact that finitely presented classical $C(7)$ small-cancellation groups are hyperbolic.