Let $G$ be a $(v,k,\lambda,\mu)$ strongly regular graph. After perusing through Brouwer's tables of parameters I have formed the conjecture $$\lambda-\mu \leq \frac{k}{2}.$$

So far I have not been able to prove it, though it seems like an easy statement. Have you seen something like this?

P.S. $\lambda-\mu \leq \frac{k}{C}$ for a universal constant $C$ would also be interesting to me.

UPDATE: I can now prove $\lambda-\mu \leq \frac{2}{3}k$ but the original question is still in force!