Forbidden subposet problems in the linear lattice

We study weak and strong forbidden subposet problems in the linear lattice $L_n(q)$. We show that the $q$-analogues of the Bukh–Griggs–Lu conjecture fail for every prime power $q$. If $q\ge 3$ and $n$ and $d$ have opposite parity, then $L{a}_q(n,L_d(q))=L{a}_q^{\ast }(n,L_d(q))={\mathrm{\Sigma }}_q(n,d)$. We also show that $L{a}_q(n,D_s)=L{a}_q^{\ast }(n,D_s)={\mathrm{\Sigma }}_q(n,2)$ for $2\le s\le q$ and all $n$, and for $s=q+1$ when $n$ is odd. For $n$ even, a maximum strong $D_{q+1}$-free family can be contained from the three middle layers. The case $s=2$ settles the $q$-analog of the diamond conjecture in the affirmative.