Almost
Specific range when it all comes down to it is TAS/sfc x total thrust.
SFC x Thrust being fuel flow.
Over time weight decreases and therefore the total drag drag curve will move down and left. This means that the IAS for the long range cruise speed will fall and the drag for this speed will fall.
Less drag will require less thrust, so total thrust can be reduced. Therefore fuel flow will fall. However, if IAS is being decreased to maintain the IAS LRC then TAS and MACH NO will fall correspondingly. This allows you to climb slowly, to benefit from reduced sfc. Climbing slowly as the IAS falls allows you to keep the TAS and MACH NO. constant and therefore keep the compressibility effect effect on drag out of the equation.
All in all, SR increases because of reduced thrust required, and if you climb to stay with the optimum altitude the point is further enforced as sfc reduces.
This was a duffers guide, so excuse the symplistics.