Should I upgrade the cable of my Line Conditioner?

No. Furman supplies a suitable cable. Replacing it will be a waste of money.

I was told that the manufacturer only supplies a power cord that is good enough to get you started until you can save enough money to buy a better power cord. Does this not hold true for the original cord that goes in the power conditioner? Wouldn’t that be the original sin? Why don’t we need to replace the wiring inside the wall?

Just replaced the wiring in my wall.  10ga to the service panel.  Recommend 10 ga power cords for amps.

Most likely would make an improvement as others have mentioned. Many times it depends upon your selection of power cords. How much ($) are you willing to spend, then perhaps you can get some suggestions.

I also use Furman Elite-15 Power Conditioners in my system. I did upgrade to better cords. For me, it made a nice difference. Although, for me - I would upgrade cords to my amp, pre-amp, CD transport and DAC first. Those had bigger bang-for-te-buck returns in my system.