It will be mostly useless.
Every electrical circuit has current flowing in a loop. When a circuit contains an antenna, there are always, in some form, two halves to the antenna, which have opposite voltages and currents (and the current loop is completed by "empty space", much like in a capacitor). When you see an antenna that looks like one wire sticking out, there is actually something else forming the other half, usually being whatever the circuit ground is (possibly a metal casing or a power supply ground).
What you have there is an unusual form of ground plane antenna, where a plane of metal (or approximation by wires) is used to form that other half. If you leave it disconnected, then you're removing it from the circuit (except for some very small amount of capacitive coupling) and the coax shield (and, in part, whatever it's attached to) will be pressed into service as the other half of the antenna.
This is undesirable because it means the antenna does not have stable characteristics (depends on metal near the coax), is not overall as far away from obstructions, and because noise from your receiver or other equipment that is conducted along the coax shield will be added to the received signal. (This last effect is not truly eliminated unless you also have a choke/unun at the feed point.)
(By the way, the instructions describe as a series of dipole antennas. Note that an ordinary dipole, with a feed point at the midpoint of the wire, does not require any sort of ground or counterpoise. This antenna has its feed point at one end of the wire.)