Csóka's weak 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 $k$ is an integer and $n\geq dk$, then

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

The uniform hiding strategy gives the corresponding upper bound, and the conjecture is known for $d=2$ when $k$ is an integer; the paper's Theorem proves it when $n$ is sufficiently large, while the stated range remains unresolved in general.