Csóka's strong caching-game value conjecture

Let $v_A(n,d,k)$ denote the value of Alpern's Caching Game with $n$ doors, $d$ treasures, and searcher power $k$. Csóka's conjecture. If $k\geq 1+\frac{2}{d-1}$ and $n\geq dk$, then

$$v_A(n,d,k)=\frac{k^d}{\binom{n+d-1}d}.$$

This strengthens Csóka's integer-$k$ conjecture, apart from small known cases. The paper notes that the threshold on $k$ is motivated by a construction, while its theorem establishes the analogous value only for sufficiently large $n$; the full stated range remains open.