  Let $F\subseteq [0,1]^2$ be a Bedford-McMullen carpet defined by exponents
$m>n$, that projects to $[0,1]$ on the $y$-axis. We show that under mild
conditions on $F$, there are many non principle lines $\ell$ such that $\dim^*
F\cap \ell = \dim^* F -1$, where $\dim^*$ is Furstenberg's star dimension
(maximal dimension of a microset). This exhibits the sharpness of recent
Furstenberg-type slicing theorems obtained by Algom (2020) about upper bounds
on the dimension of every such slice.
