Conjecture on the six smallest inversion numbers

For a fixed positive integer $b$, let $\operatorname{inv}(a,b)$ denote the inversion number and let $I_n(b)$ be its $n$th smallest value, where $1\leq n\leq6$. Six-smallest-values conjecture. For $1\leq n\leq6$, one has

$$I_n(b)\geq\frac{(n-1)(b+1)(b+1-n)}{4n},$$

with equality

$$\operatorname{inv}(n,b)=\frac{(n-1)(b+1)(b+1-n)}{4n}$$

when $b\equiv-1\pmod n$. This extends the proved bounds for the second smallest and second largest values to a conjectural description of the six smallest inversion numbers; the source does not indicate that the full statement has been resolved.