The threshold conjecture for star decompositions of random regular graphs

Let $d\geq 3$ be fixed, and let $\kindactual{d}$ denote the largest positive integer $k$ such that, asymptotically almost surely, the random $d$-regular graph $\mathcal{G}_{n,d}$ contains an independent set of size $\frac{(2k-d)n}{2k}$. For fixed positive integers $d,k$ with $d\geq 3$, star-decomposition threshold conjecture.

$$\mathop{\mathrm{Pr}}\big(\mathcal{G}_{n,d} \text{ has a $k$-star decomposition}\big)\rightarrow \begin{cases} 1 & \text{ if $k \leq \kindactual{d}$,}\\ 0 & \text{ if $k > \kindactual{d}$.} \end{cases}$$

The independent-set condition is necessary because, when $k>d/2$, the leaves in a $k$-star decomposition form an independent set of size $\frac{(2k-d)n}{2k}$. The conjecture asserts that this necessary condition determines the asymptotic threshold for $k$-star decompositions of random regular graphs; the supplied text does not state that it has been proved or disproved.