The theoretical answer required by the ATPL may well be different to the reality, so here goes.
Depending on the weight of the aircraft, the ROD may increase slightly as the IAS starts to reduce. It's so slight though as not be to readily noticeable. Working against it is that the ROD is always trending downwards anyway as you climb into thinner air. Any increase in ROD may also be down to the autopilot chasing the speed as it finds it harder to hold it accurately in the thin air. This phenomenon will in fact mask any increase due to mach conversion. Add in varying temperatures will alter the Mach number, thus further causing the autopilot to chase the speed when in the mach regime.
This is for A320. Typical climb is 300kt to conversion (about FL290 depending on temperature) then M0.78. The IAS will then decrease to about 260kt in the cruise.