The exact growth conjecture for the three-container pouring function

Let $h(N,3)$ denote the maximum capacity of an initial state with three containers that requires exactly $N$ pourings to solve the generalized double pouring problem.

Exact growth conjecture. For each positive integer $N$, one has

$$h(N,3)=5\cdot 2^{N-1}.$$

Computational results motivate this exact formula, while Frei et al. have shown that infinitely many initial states satisfy the asserted bound. Whether the equality holds for every positive integer $N$ remains open.