Just for clarification, what do you mean by the horizontal distribution? Are you asking about the distribution of the real parts of the zeros of $\zeta'(z):=\frac{\operatorname{d}}{\operatorname{d}z}\zeta(z)$? In that case, the answer is no. It is known that $\zeta'(z)$ does not satisfy the Riemann hypothesis, due to interaction with the trivial zeros, so the distribution of the real parts of these zeros is an interesting question, but not one on which I can comment. Our work only deals with the derivatives of the symmetric zeta function $\Lambda(z)$ which are all expected to satisfy both the Riemann hypothesis and the GUE model. If your question was about the distribution of these zeros, then yes, our theorem suggests that the low-lying zeros follow the GUE model with increasing accuracy as the order of the derivative increases.

Just for clarification, what do you mean by the horizontal distribution? Are you asking about the distribution of the real parts of the zeros of $\zeta'(z):=\frac{\operatorname{d}}{\operatorname{d}s}\zeta(s)$? In that case, the answer is no. It is known that $\zeta'(s)$ does not satisfy the Riemann hypothesis, due to interaction with the trivial zeros, so the distribution of the real parts of these zeros is an interesting question, but not one on which I can comment. Our work only deals with the derivatives of the symmetric version of the zeta function, $\Lambda(z)$, which are all expected to satisfy both the Riemann hypothesis and the GUE model. If your question was about the distribution of these zeros, then yes, our theorem suggests that the low-lying zeros follow the GUE model with increasing accuracy as the order of the derivative increases.