I do not know what defines the optimum LRC speed for each aircraft, aeronautically speaking. I suspect it has something to do with optimum AoA but even then I am not sure if the LRC speed should increase or decrease as the GW is being reduced during the flight.
Q: "I do not know what defines the optimum LRC speed for each aircraft"
The standard aviation industry definition is that LRC [Long Range Cruise] is the speed which offers 99% of the maximum possible fuel mileage [...] LRC has long served as an accepted compromise between a desire for higher fuel mileage and a desire for shorter cruise times. No, it’s not the best fuel mileage, but neither is it the slowest cruise speed.
— Boeing Jet Transport Performance Methods. ch. 32 (MRC and LRC), p. 11.
Q: "Does the optimum LRC speed increase or decrease as the GW of the aircraft decreases during a long range flight?"
"As the weight of the aircraft decreases in-flight," the LRC (Long Range Cruise) Mach number decreases, because the L/D curve shifts up and to the left with weight reduction and so would the fuel mileage curve, because a slower speed is needed for the same optimum angle of attack:
Note: typically in jets with an FMC, unless having to fly a fixed Mach number for e.g. oceanic separation, the optimum Mach number is automatically adjusted as fuel is burned, whether for LRC, MRC, or a certain cost index.