There are two energy usage issues at play here.
1st; energy consumed on a constant torque load through a VFD is directly proportional to speed, as is HP. So IF your load can use less HP to get the job done, a VFD can facilitate that. But is should be pointed out that any other mechanical method of reducing the speed will do exactly the same thing. So for instance, if you use a gearbox or variable sheave drive to lower the workload on the motor, that too will save energy (let's ignore the inefficiencies of varidrives for this argument), as long as the work being done by the motor is less than it was before. For that matter, if you can live with less agitation, you can also achieve the same effect by varying the duty cycle (on-off control) of the agitator. So that means it isn't that a VFD will not save energy by reducing the speed, but that the amount of energy it saves will not be significantly different than any other method of reducing the workload over time.
2nd; the energy savings involved in variable torque loads is different. By using the VFD, again on a load that CAN be varied such as flow control in a centrifugal pump or fan, you save EXTRA energy when compared to other flow control methods by taking advantage of what are called the "affinity laws". When you choke off the flow of a constant speed pump, you do still reduce energy consumption because you reduce the work being done, but when you lower the speed of the motor instead of choking off the flow, you save MORE energy. The difference between the two levels represents the energy savings that we all get so excited about with VFDs. It can be significant.
So to summarize, you can save energy in a constant torque motor application by using a VFD, but no more than you could using far less expensive or less complicated technologies. The extra energy savings you get with variable torque loads is real, but does not translate to constant torque applications.