Soundararajan's conjecture on zeros of the zeta-function derivative and gaps between zeta zeros

Throughout, let $s=\sigma+it$ be a complex variable. Write $\rho=\beta+i\gamma$ and $\rho'=\beta'+i\gamma'$ for generic zeros of $\zeta(s)$ and $\zeta'(s)$, respectively. If $\zeta(1/2+i\gamma)=0$, let $\gamma^+$ be the smallest $t>\gamma$ for which $\zeta(1/2+it)=0$. Assume the Riemann Hypothesis.

Soundararajan's conjecture. The following two statements are equivalent:

$$\textnormal{(A)}\quad \liminf_{\gamma'\rightarrow\infty}(\beta'-1/2)\log\gamma'=0;$$ $$\textnormal{(B)}\quad \liminf_{\gamma\rightarrow\infty}(\gamma^+-\gamma)\log\gamma=0.$$

Both assertions are expected to be true, but the conjecture seeks a direct relation between them without relying on their individual validity. One direction, that (B) implies (A) on the Riemann Hypothesis, was known in the paper's discussion; the converse remained unresolved there.