Find a synchronizing automaton with reset threshold exceeding (n-1)^2, or extend Cerny verification to n=13

A deterministic finite automaton (DFA) with state set $Q$ ($|Q|=n$) over an input alphabet is synchronizing if some input word $w$ (a reset word) maps every state to a single state: $|Q\cdot w|=1$. The reset threshold is the length of a shortest reset word. Cerny's conjecture (1964) asserts that every synchronizing $n$-state DFA has reset threshold at most $(n-1)^2$; the Cerny automata achieve exactly $(n-1)^2$. The conjecture is verified by exhaustive search only for small $n$: all 2-letter automata with $n \le 12$ states, and all 3-letter automata with $n \le 8$ states, satisfy it. Exhibit any synchronizing DFA whose reset threshold is strictly greater than $(n-1)^2$ (a counterexample, refuting the conjecture), or extend the exhaustive verification to all 2-letter automata on 13 states.

Acceptance. FULLY RESOLVES (refutation): a synchronizing DFA - given by its transition function (an $n\times k$ table) - together with a proof that its reset threshold exceeds $(n-1)^2$. The reset threshold is machine-checkable via breadth-first search over subsets of $Q$ from the full set $Q$ to a singleton (the power automaton), computing the exact shortest reset word length; a certificate is the BFS distance plus the resulting shortest word. FULLY (verification extension): an exhaustive check that all synchronizing 2-letter 13-state DFAs have reset threshold $\le 144=(13-1)^2$, with reproducible search code and census counts. PARTIAL: a family of automata with reset thresholds close to $(n-1)^2$ not previously recorded, or verification of a nontrivial restricted class at $n=13$. Provide the transition table and a reset-threshold verifier.