  We construct the first known infinite family of quasi-isometry classes of
subgroups of hyperbolic groups which are not hyperbolic and are of type
$\mathrm{FP}(\mathbb{Q})$. We give a simple criterion for producing many
non-hyperbolic subgroups of hyperbolic groups with strong finiteness
properties. We also observe that local hyperbolicity and algebraic fibring are
mutually exclusive in higher dimensions.
