Let $p$ and $q$ be distinct projections in a von Neumann algebra $M$. If $\|p-q\|＜1$, then there exits a unitary $u$ such that $q=upu^*$ and $\|1-u\|＜\sqrt{2}\|p-q\|$.
If we only know that $p$ is unitarily equivalent to $q$, can we deduce that there exits a unitary $u$ such that $q=upu^*$ and $\|1-u\|＜\sqrt{2}\|p-q\|$?
Let $$ p=\begin{bmatrix}1&0\\0&0\end{bmatrix},\qquad\qquad q=\begin{bmatrix}0&0\\0&1\end{bmatrix},\qquad\qquad u=\begin{bmatrix}0&-1\\-1&0\end{bmatrix}. $$ Then $upu^*=q$, and $$ \|1-u\|=2>\sqrt2=\sqrt2\,\|p-q\|. $$ Any other unitary with $upu^*=q$ is of the form $$ v=\begin{bmatrix}0&b\\c&0\end{bmatrix}, $$ with $|b|=|c|=1$. Then $$ \|1-v\|=\Bigg\|\begin{bmatrix}1&-b\\-c&1\end{bmatrix}\Bigg\|\geq\sqrt2. $$