co.combinatorics - Two [n] to [n] function families

$bf Note.$ This question had a bounty, so at the end I accepted the best (and only) answer but in fact it is still open. It is (hopefully) equivalent to this question, if you have any ideas, please post them there.

$bf Question.$
Fix n. We are interested in the biggest t for which there exist two families of functions, $P_i,Q_i$, of size t from [n] to [n] such that for any $i,j$ whenever we consider the infinite sequence $P_i(Q_j(P_i(Q_jldots P_i(3))ldots)$ (where the number of iterations tends to infinity), it contains no 2's and infinitely many 1's if $i=j$ and it contains no 1's and infinitely many 2's if $ine j$.

$bf A lower bound.$
I know a construction that shows that $tge 2^{frac n2-O(1)}.$ For every subset $S$ of [n] that contains exactly one of $2k$ and $2k+1$ for $2le kle frac n2-2$ we construct a pair of functions, $P_S,Q_S$ as follows. For any number m denote by $m^+$ the smallest element of $S$ that is bigger than m or if all elements of $S$ are at most m then define it to be 1.
$P_S(1)=1, P_S(2)=2$ and for bigger $m$'s $P_S(m)=m^+$, while $Q_S(1)=1, Q_S(2)=2$ and for bigger $m$'s $Q_S(m)=m$ if $min S$ and $Q_S(m)=2$ if $mnotin S$. This way we go through all the elements of S and end in 1 if the functions have the same index, but we are pushed to 2 if they differ.

$bf Upper bound.$
It is of course true that $tle n^n$. So can you do better than $2^n$?