So, it's known that PCP is undecidable even when we fix the number of tiles to $n \geq 7$.
I'm wondering, can anything similar be said for when there is a fixed word length?
To be precise, here's the problem:
Given fixed $m$ and $n$, with $n \geq 7$, and words $u_1, \ldots u_n$ and $v_1 \ldots v_n$ such that $|u_i| \leq m$ and $|v_i| \leq m$, is there an index sequence $i_1, \ldots i_k$ such that $u_{i_1} \cdots u_{i_k} = v_{i_1} \cdots v_{i_k}$.
For what values of $m$, if any, is this known to be undecidable?
Note that this is similar to this question, but none of the 8 linked papers seemed by their titles to answer my question, and I haven't fully read all 8 of them yet.