How far you go will depend on your audio system upgrade aspirations. If possible. it would be good to get the grounding sorted while you can too, but be prepared for revealing sound reproduction issues you have not experienced before.
If you want to go the full distance, install a dedicated grounding system with copper tape and rods drilled and grouted into bedrock, outside your room. Place Bentonite clay beneath the mat and keep damp to maintain low resitivity. For peace of mind, get your electrician to do a resistivity test. The lower the Ohm number, the better it is conducting to ground. Appreciate that this may not be physically and financially justified now, but you may wish you had done it later (???)