As we know, breaking a Transposition Cipher is quite trivial and can be easily done by either exploiting single letter frequencies (not $d$-gram frequencies though, given that $2 < d < L$), or by employing a simple CPA, thus deriving the snippets length $L$ and the transposition matrix (key).
If we imagine to encrypt the plaintext twice, employing two different keys, would this help improving security?
If we employ the same snippet length $L_1 = L_2$, this clearly won't improve security as an attacker could easily find the key $k_3 = k_1 \times k_2$ (where $\times$ is the matrix product), without even bothering about $k_1$ and $k_2$.
However, that's not the case if $L_1 \neq L_2$. So I guess there's some kind of improvement; can you show me if and how this is true?