= Solution
\b[c1 is true for density-state controllability.] Every qubit <density operator> can be written $\rho=\lambda_2I+(\lambda_1-\lambda_2)|\psi\rangle\langle\psi|$ using its <eigenvalues>. A control taking one pure <orthogonal projector> to any other therefore also connects every pair of qubit <density operators> with the same <spectrum>. In Lie-algebra language, $\mathfrak{sp}(1)=\mathfrak{su}(2)$; there is no larger-dimensional symplectic exception. This does not supply arbitrary <global phase>: $SU(2)$ alone is already a counterexample to an implication involving strict operator controllability.
\b[c2 is false.] For every even $N=2\ell\ge4$, the defining $Sp(\ell)$ action is transitive on unit vectors, so a system generating this algebra is pure-state controllable. It still preserves additional mixed-state invariants and lacks <density operator controllability>, as the preceding explicit four-dimensional example demonstrates. <pure states> have fewer orbit requirements than general <mixed states>.
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