Hull-dimension monotonicity hypothesis for binary and ternary linear codes

Let $B_{n,k,\ell,q}$ be the number of inequivalent linear codes of length $n$, dimension $k$, and hull dimension $\ell$ over $\mathbb{F}_q$. Hull-dimension monotonicity hypothesis. If $q=2$ or $3$ and $n\ge 2k$, then

$$\min\{B_{n,k,0,q},B_{n,k,1,q}\}>B_{n,k,2,q}>\cdots>B_{n,k,k,q}.$$

This hypothesis is motivated by classification tables and computational results for binary and ternary linear codes, but the stated inequality is not established in the supplied text.