  Let $F$ be a Bedford-McMullen carpet defined by independent integer
exponents. We prove that for every line $\ell \subseteq \mathbb{R}^2$ not
parallel to the major axes, $$ \dim_H (\ell \cap F) \leq \max \left\lbrace 0,\,
\frac{\dim_H F}{\dim^* F} \cdot (\dim^* F-1) \right\rbrace$$ and $$ \dim_P
(\ell \cap F) \leq \max \left\lbrace 0,\, \frac{\dim_P F}{\dim^* F} \cdot
(\dim^* F-1) \right\rbrace$$ where $\dim^*$ is Furstenberg's star dimension
(maximal dimension of microsets). This improves the state of art results on
Furstenberg type slicing Theorems for affine invariant carpets.
