K-semistability of smooth Fano threefolds with Picard rank one

Let $X$ be a smooth Fano threefold, and write $\rho(X)$ for its Picard rank.

K-semistability conjecture. Every smooth Fano threefold with $\rho(X)=1$ is K-semistable.

The conjecture is motivated by the classification of smooth Fano threefolds and the known K-polystability or K-stability of the families with Fano index at least two. The source presents it as a speculation for the remaining Picard-rank-one families.