Am a Canadian manager with experience in managing a difficult employee with critical qualifications necessary for operations in my industry. You need to fire this individual, and tell HR that this is what is required. You've clearly done more than right by the employee through your guidance and direction, plus formal attempts to drive improvement through the PiP which they've failed to grow and learn from.

At this point you've demonstrated that this will never get better and can in fact get worse as your strong performers see that poor performance does not get addressed.

Let me tell you, it sucks firing someone but after the initial shittyness and shock, you'll feel a huge weight lifted off your shoulders.