zeros of exponential polynomials

Let $\exp[n;z]$ denote the $n$th Taylor polynomial for the exponential function.

In the 1920's Szegő initiated the study of the asymptotic properties of the zeros (rescaled by dividing by $n$) of this family of polynomials and one consequence of his results is that they can approach arbitrarily closely to the imaginary axis. This prompts the following question:

Is it possible for $\exp[n;z]$ to have a root which lies precisely on the imaginary axis?

This is equivalent to asking if there is a simultaneous real zero of the two polynomials $\cos[n,z]$ and $\sin[n,z]$. But for any $n$, one of these two polynomials is the derivative of the other, so they are only simultaneously zero at a repeated root of the higher-degree one.

So the question is equivalent to asking if the Taylor polynomial centered at 0 of $\sin$ or $\cos$ ever has a repeated real root.

Edit: Apparently this blog has been following our discussion and states that their are no repeated roots.