Proof of CFL condition for RKDG scheme

The cfl condition for linear advection equation $$u_t + a u_x = 0$$ using a DG method of degree $k$ polynomials, upwind flux and an RK scheme of $k+1$ stage/accuracy is stated to be $\frac{1}{2k+1}$ for $L^2$ stability. However I have not been able to find a paper which proves this though it is mentioned in some papers that the proof exists for k=0,1,2. Please point out any existing literature on this.

We have derived a closed form expressions for the eigenvalues of the DG spatial discretization applied to the onedimensional linear advection equation with periodic boundary conditions and the upwind flux. [...] We have also shown that $(p + 1)(p + 2)$ is a guaranteed bound on the size of the eigenvalues which can be used to compute the CFL condition for large p. However, we have also proved that the growth rate of the largest eigenvalue is less than $(p + 1)^2$. We conjecture that a more accurate rate is proportional to $(p + 1)^{1.75}$. This is in contrast with the currently assumed quadratic rate for the DGM [13] and various spectral methods [12].