A maintenance drone on a skyscraper drops two calibration weights into a vertical service conduit. Weight $\alpha$ is released from rest at a height of $h$ metres above the conduit's base. Weight $\beta$ is released from rest $t_{lag}$ seconds later from a point $\rho h$ metres above the base, where $0<\rho <1$. It is observed that both weights strike the base of the conduit exactly 9 seconds after weight $\alpha$ was first released. Assuming air resistance is negligible, prove that:

$$t_{lag}=9(1-\sqrt{\rho })$$

Ensure every step of your derivation is clearly justified.