The non-cylindricity conjecture for Fano threefolds of genus 7

Let $X$ be a smooth Fano threefold such that $\rho(X)=1$, $\iota(X)=1$, and $\operatorname{g}(X)=7$. The non-cylindricity conjecture. Then $X$ is not cylindrical. The source states that no examples of cylindrical smooth Fano threefolds with Picard rank $1$ and genus $7$ are known and believes that every such threefold is non-cylindrical; the assertion remains open.