The recursive upper-bound conjecture for multidimensional egg-drop problems

Let $d$ be a positive integer, let $N_1,N_2,\dots,N_d$ be the side lengths of a $d$-dimensional setting, and let $k$ be the number of eggs. Write $P_d(k)$ for the minimum number of drops required in the worst-case scenario under the strategy considered in the paper.

Recursive upper-bound conjecture. In a $d$-dimensional setting,

$$P_d(k)\leq\left\lceil (k-d+1)\cdot(N_1+N_2+\dots+N_d)^{1/(k-d+1)}\right\rceil,$$

for $k\geq d$.

The conjecture extrapolates the recursive upper bounds established in the one-, two-, and three-dimensional cases to arbitrary dimension. It concerns the performance of the same critical-point strategy in the worst case; the general $d$-dimensional claim is presented as suggested by the lower-dimensional pattern, with no resolution supplied here.