Conjecture on derivative-zero density and small zeta-zero gaps

Assume the Riemann hypothesis and define the normalized gap and derivative-zero quantities by

$$\lambda_j=(\gamma_{j+1}-\gamma_j)\log\gamma_j,\qquad \lambda_j'=(\beta_j'-\tfrac12)\log\gamma_j'.$$

For $\nu>0$, let

$$m(\nu)=\liminf_{J\to\infty}\frac{1}{J}\#\{j\leq J:\lambda_j\leq\nu\},\qquad m'(\nu)=\liminf_{J\to\infty}\frac{1}{J}\#\{j\leq J:\lambda_j'\leq\nu\}.$$

Derivative-zero density conjecture. If $m'(\nu)\gg\nu^\alpha$ for some $\alpha<2$, then $m(\nu)>0$ for all $\nu>0$. This is proposed as a refinement of Soundararajan's conjecture. It asserts that sufficiently substantial density of zeros of $\zeta'$ close to the critical line forces positive density of zeta zeros with every positive normalized gap threshold; the paper does not state that it has been proved.