PCP when upper and lower words have different length
The problem doesn't become any easier.
Suppose we are given an instance $a_1,\ldots,a_n,b_1,\ldots,b_m$ of PCP.
Let $\sigma$ be a new symbol. Add to the first list the words $a'_1 = \sigma a_1,\ldots,a'_n = \sigma a_n$, and to the second list the word $b'_0 = \sigma$.
If the new PCP instance has a solution, then we can obtain a solution of the original PCP instance by removing all occurrences of $\sigma$. Now suppose that the original PCP instance has a solution
$$ a_{i_1} \ldots a_{i_s} = b_{j_1} \ldots b_{j_t}. $$
If $s \neq t$ then this is a solution of the new PCP instance in which $s \neq t$. If $s = t$, then consider the following solution instead:
$$ a'_{i_1} \ldots a_{i_s} = b'_0 b_{j_1} \ldots b_{j_t}. $$