Borozan et al.'s conjecture on k-proper partitions

Let $G$ be a graph of order $n$, and let $k$ be an integer at least $2$. A partition $\mathcal{P}$ of $V(G)$ is $k$-proper if every part $P\in\mathcal{P}$ induces a $k$-connected subgraph of $G$, where a graph is $k$-connected if it has more than $k$ vertices and deleting any set of fewer than $k$ vertices leaves it connected. Write $\delta(G)$ for the minimum degree of $G$. Borozan et al.'s conjecture. If

$$\delta(G)\geq\sqrt{(k-1)n},$$

then $G$ has a $k$-proper partition $\mathcal{P}$ satisfying

$$|\mathcal{P}|\leq\frac{n-k+1}{\delta(G)-k+2}.$$

This would improve the known bound with constant $c=\frac{2123}{180}$ in the corresponding minimum-degree theorem, reducing that constant to $1$ and sharpening the bound on the number of parts.