Boruzanli and Gauci's super-connectivity conjecture for Kneser graphs

Let $KG(n,k)$ be the graph whose vertices are the $k$-subsets of $[n]=\{1,\ldots,n\}$, with two vertices adjacent exactly when the corresponding subsets are disjoint. A vertex cut is a set of vertices whose deletion disconnects the graph; the super-connectivity $\kappa_1$ is the size of a smallest nontrivial vertex cut, meaning one that does not isolate a single vertex. Boruzanli and Gauci's conjecture. If $n\geq 2k+1$, then the super-connectivity $\kappa_1$ of $KG(n,k)$ is

$$\kappa_1=\begin{cases} 2\left({n-k \choose k}-1\right),&2k+1\leq n<3k,\\ 2\left({n-k \choose k}-1\right)-{n-2k\choose k},&n\geq 3k. \end{cases}$$

The conjecture gives the expected minimum nontrivial vertex-cut size for Kneser graphs, refining the known connectivity result. The supplied source does not state whether it has been proved or disproved, so its status remains open.