Equality and strict upper-bound conjecture for the estimates of direct powers of free distributive lattices

Let $g_3^{\ast}(n)$, $g_3^{\ast\ast}(n)$, and $g_r(n)$ be the functions defined earlier in the paper, and let $\mathbb{N}^{\geq k}=\{n\in\mathbb{N}:n\geq k\}$. Equality and strict upper-bound conjecture. We guess that

$$g_{3}^{\ast}(n)=g_{3}^{\ast\ast}(n)$$

for all $n\in\mathbb{N}^{\geq 3}$, and that

$$g_{3}^{\ast\ast}(n)<g_r(n)$$

for all $n\in\mathbb{N}^{\geq 6}$. This conjecture concerns the comparison of the paper's computable estimates for the minimum size of generating sets of direct powers of free distributive lattices; the stated theorem establishes the corresponding equality only for $n\in\{3,4,\dots,300\}$ and the strict inequality only for $n\in\{5,6,\dots,300\}$, so the extensions to all indicated $n$ remain open.