The GUE random matrix model predicts that the zeroes should satisfy the local statistics of random matrices. It doesn't predict that the zeroes should satisfy the global statistics of random matrices, because it's not clear what that would even mean unless the zeroes are all contained in some bounded interval. (In fact they get more frequent the further away from the origin).

The Hermite polynomials satisfy the global statistics of random matrices (the semicircular law) but not the local statistics (their zero spacing is very even). So it is not clear how a precise relationship between the GUE conjecture and this new theorem could be formulated.

For a toy model of what's going on here, we could take the function field model, where zeta functions look like $$ \frac{ \sum_{n=0}^{2g} a_n q^{-ns} }{ (1-q^{-s}) (1-q^{1-s} ) }.$$ The first step here is to multiply by a factor to make the functional equation as simple as possible, which for us is $q^{gs}$, and the second step is to multiply by a factor that kills the poles, which for us is $ (1-q^{-s}) (1-q^{1-s} )$. So the object being differentiated is $$\sum_{n=0}^{2g} a_n q^{(g-n) s}.$$ The $k$th derivative (with respect to $s$) is $$ (\log q)^k \sum_{n=0}^{2g} a_n (g-n)^k q^{g-ns}.$$ Renormalizing, we get $$ \sum_{n=0}^{2g} a_n \left(1 - \frac{n}{g} \right)^k q^{g-ns}.$$ As $k$ goes to $\infty$ and remains even, this converges to $$ a_0 q^{gs} + a_{2g} q^{-2g s},$$ which has all roots on the half-line, perfectly evenly spaced, since $$a_{2g} = q^g a_0$$ by the functional equation.

If we view these roots as lying on a circle (i.e. we take $q^{-s}$ for $s$ a root), we can say that they perfectly satisfy the global GUE statistics, being evenly distributed on the circle, but do not satisfy global GUE statistics, being perfectly evenly spaced. This is true regardless of whether our original zeta function behaved like a characteristic polynomial of a random matrix, or even whether it satisfied the Riemann hypothesis.

It is possible that a similar phenomenon is occurring for the Jensen polynomials of high derivatives.

The GUE random matrix model predicts that the zeroes should satisfy the local statistics of random matrices. It doesn't predict that the zeroes should satisfy the global statistics of random matrices, because it's not clear what that would even mean unless the zeroes are all contained in some bounded interval. (In fact they get more frequent the further away from the origin).

The Hermite polynomials satisfy the global statistics of random matrices (the semicircular law) but not the local statistics (their zero spacing is very even). So it is not clear how a precise relationship between the GUE conjecture and this new theorem could be formulated.

For a toy model of what's going on here, we could take the function field model, where zeta functions look like $$ \frac{ \sum_{n=0}^{2g} a_n q^{-ns} }{ (1-q^{-s}) (1-q^{1-s} ) }.$$ The first step here is to multiply by a factor to make the functional equation as simple as possible, which for us is $q^{gs}$, and the second step is to multiply by a factor that kills the poles, which for us is $ (1-q^{-s}) (1-q^{1-s} )$. So the object being differentiated is $$\sum_{n=0}^{2g} a_n q^{(g-n) s}.$$ The $k$th derivative (with respect to $s$) is $$ (\log q)^k \sum_{n=0}^{2g} a_n (g-n)^k q^{g-ns}.$$ Renormalizing, we get $$ \sum_{n=0}^{2g} a_n \left(1 - \frac{n}{g} \right)^k q^{g-ns}.$$ As $k$ goes to $\infty$ and remains even, this converges to $$ a_0 q^{gs} + a_{2g} q^{-2g s},$$ which has all roots on the half-line, perfectly evenly spaced, since $$a_{2g} = q^g a_0$$ by the functional equation.

If we view these roots as lying on a circle (i.e. we take $q^{-s}$ for $s$ a root), we can say that they perfectly satisfy the global GUE statistics, being evenly distributed on the circle, but do not satisfy local GUE statistics, being perfectly evenly spaced. This is true regardless of whether our original zeta function behaved like a characteristic polynomial of a random matrix, or even whether it satisfied the Riemann hypothesis.

It is possible that a similar phenomenon is occurring for the Jensen polynomials of high derivatives.