As I said, only 2 pairs of the 4 are in use if speed is 10 or 100Mbit
other two pairs? not in use, they could be broken/cut/open and cable would still work at that speed.
In this case, pins 1, 2, 3 and 6 are in use (orange and green pairs) and rest are not.
Of course that assumes that the network speed is at 10Mbit or 100Mbit and not 1000Mbit
If the speed is at gigabit and tester claims it's open.. then something is more fishy since gigabit should not work with open wires.
The tester tests all 4 pairs (8 wires) and complains if any of them fail, as it should.
Also if router says "not connected" is it through said tester (based on software, software might draw it's own conclusions as it wants, in this case assuming that missing 1 or 2 pairs=not connected) or based on the link light?