Ross and Thomas developed slope-stability of $(X,L)$ where $X$ is an $L$-polarised variety and $L$ is an ample line bundle, as an obstruction to K-stability of $(X,L)$.

DISCLAIMER: (Forgive me if I don't define what these are, but for the purpose of the question if you do not know them well already is not going to help you. Moreover the definition is rather technical. Also, I am not an expert in these topics, so I expect to say a couple of things wrong without being aware of it)

K-stability is an obstruction to the existence of Kahler-Einstein metrics on $X$. Therefore slope-stability can be seen as a tool to decide when a Fano variety does not admit a Kahler-Einstein metric:

Kahler einstein $\Rightarrow$ K-stable $\Rightarrow$ slope-stable.

The last arrow is not strict, i.e. there are slope-stable $(X,L)$ which are not K-stable. On the other hand, computing slope-stability is much easier than computing K-stability.

I know that K-stability has other applications, for instance if $(X,\mathcal{O}(-mK_X)),\ m\in \mathbb{Z}_{>0}$ satisfies certain conditions (including being Fano) and K-semistability, then the singularities of $X$ are log terminal by a result by Odaka in Annals. Therefore K-stability is interesting not only within the Kahler-Einstein problem.

I was wondering if there is any other applications of slope-stability other than as an obstruction to K-stability.

(answers considering the log setting are also welcomed)