Differentiation is better than midpoint

Let $f$ be an entire function of order $1$ that is real on the real axis and has only real zeros, and list its zeros as $z_j$ in increasing order. If $q<p$ are consecutive zeros of $f'$, then

$$\inf_n \frac{1}{2}(z_{n+2}-z_n)\le p-q\le \sup_n \frac{1}{2}(z_{n+2}-z_n).$$

Differentiation is better than midpoint conjecture. The gaps between consecutive zeros of $f'$ lie between the infimum and supremum of the corresponding two-step midpoint-process gaps of the zeros of $f$.

This would formalize the idea that differentiation evens out zero spacings at least as effectively as the midpoint process. The supplied text presents it as a proposed result and gives no resolution, so its status is open.