The type-independence conjecture for $v$-palindromes

Let $m$ be a $v$-palindrome with two representations

$$m=n_1(k_1)=n_2(k_2).$$

For each representation, define the type of $m$ with respect to the corresponding $n_i$ as in the source.

Type-independence conjecture. The type of $m$ with respect to $n_1$ is the same as the type of $m$ with respect to $n_2$.

The conjecture asks whether the notion of type depends only on the $v$-palindrome $m$, rather than on the chosen representation or on $n$. The supplied text gives examples supporting it but no proof or disproof.