The Banks–Martin conjecture analogue for primitive sets over function fields

Let $q\geq 5$ be a prime power, and let $\mathcal{I}_{k,q}$ denote the relevant family of primitive sets of size $k$ over $\mathbb{F}_q[x]$. Write $\mathcal{F}(\mathcal{I}_{k,q})$ for the quantity associated with this family. The conjecture asserts that

Banks–Martin conjecture analogue. For each $q\geq 5$, the inequalities

$$\mathcal{F}(\mathcal{I}_{1,q}) > \mathcal{F}(\mathcal{I}_{2,q}) > \ldots > \mathcal{F}(\mathcal{I}_{k,q}) > \mathcal{F}(\mathcal{I}_{k+1,q}) > \ldots$$

hold for all positive integers $k$.

Numerical computations support the conjecture, including the case $q=5$ for all $k<100$, and suggest that these values decrease to $1$ as $k\to\infty$.