If the object has a velocity in the direction of the singularity, it will certainly reach it in it's own reference frame. The singularity is stationary with respect to the incomming object. It's radius is decreasing but it is limited by it's size.

Say I'm moving at 1 m/s towards the singularity 1 m away. Now it's still a big singularity say 1 micrometer in radius. For the surface to be elusive, it would have to contract at at least 1 m/s for 1 sec which after 1 microsecond would diminish it to nothing.

This is a paradox formulated in antiquity except I believe the case used was a lion and a gladiator.