I was in an argument about aerodynamics and we eventually derived an equation as follows.
So I'm going by this equation:
$$RoC = \frac{\eta_p P-VD}{W}$$
Where $\eta$ represents propeller efficiency.
Assuming low Mach number:
$$TAS = IAS \cdot \sqrt{\frac{\rho_0}{\rho}}$$
$$ \text{Drag} = k(TAS)^2 \rho = k(IAS)^2 \rho_0$$
$$ \text{Specific Power} := P_s = \frac{\eta_p P}{W}$$
$$ \text{Drag Power} := P_d = \frac{TAS \cdot D}{W} = k \frac{\rho (TAS)^3}{W} = k \rho_0 \sqrt{\frac{\rho_0}{\rho}} \frac{(IAS)^3}{W}$$
The optimum climb IAS is given by:
$$\frac{dP_s}{d(IAS)}= \frac{dP_d}{d(IAS)}$$
Without loss of generality:
$$\frac{d\eta_p}{d(IAS)} = k \frac{(IAS)^2}{\sqrt{\rho}}$$
So does optimum IAS for climb vary with altitude?