If you start with a partition of $n$ in which each part has size $1$ or $2$ and take its conjugate, you get a partition of $n$ into one or two parts. The only time you get just one part, however, is when you started with the partition of $n$ into $n$ parts of size $1$. And the only time you get two identical parts is when $n$ is even, and you start with the partition of $n$ into $n/2$ parts of size $2$.
How can you add three dots to the Ferrers diagram of the original partition in such a way that the conjugate has two distinct dots, and the original partition is uniquely recoverable from the conjugate?
If $\lambda$ is the original partition, $\mu$ is the partition of $n+3$ after you’ve added the $3$ dots, and $\mu'$ is the conjugate of $\mu$, you must make sure that $\mu$ has only parts of size $1$ and $2$ and has at least one part of size $2$; that will ensure that $\mu'$ has two parts. How can you add the $3$ dots to ensure that $\mu'$ has two distinct parts? What does that say about $\mu$?