= Solution
\b[False.] At <eigenvalue> $2$, algebraic multiplicity two and minimal-polynomial exponent two force one block $J_2(2)$. At <eigenvalue> $1$, the blocks have total size five and largest size two, so there are only two possible partitions,
$$
2+2+1\qquad\hbox{or}\qquad2+1+1+1.
$$
Hence there are exactly two conjugacy classes satisfying the given <polynomial> data, represented by
$$
J_2(2)\oplus J_2(1)\oplus J_2(1)\oplus J_1(1)
$$
and
$$
J_2(2)\oplus J_2(1)\oplus J_1(1)\oplus J_1(1)\oplus J_1(1).
$$
There cannot be three mutually non-conjugate examples.
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