Let us first review the usual construction of a Mukai flop. Suppose $M$ is a smooth $2m$-dimensional projective variety over $\mathbb C$ containing a closed $m$-dimensional subvariety $W$, and suppose we have a projective birational morphism $\nu:M\to\overline M$ such that $\nu$ is an isomorphism away from $W$, and such that the restriction of $\nu$ to $W$ is a projective bundle $W\cong \mathbb P(\mathcal V)\to Y$, where $\mathcal V$ is a vector bundle on a smooth subvariety $Y\subset \overline M$. Assume further that the normal bundle $N_{W/M}\to W$ restricts to the cotangent bundle on each fibre of this projective bundle.

We can then perform the Mukai flop of $M$ along $W$ as follows: let $\widetilde M\to M$ be the blowup of $M$ along $W$. This has exceptional divisor $E = \mathbb P(N_{W/M})$. By the assumption on the normal bundle made above, using the Euler sequence we have an embedding

$$E\subset \mathbb P(\mathcal V)\times \mathbb P(\mathcal V^*)$$ which is described on each fibre over $y\in Y$ as the incidence variety

$$\{(L,H)\in \mathbb P(\mathcal V_y)\times \mathbb P(\mathcal V_y^*): L\subset H\}.$$ It can then be shown that there exists a variety $M'$ and a birational morphism $\widetilde M\to M'$ which is an isomorphism away from $E$ and whose restriction to $E$ is the projection onto $\mathbb P(\mathcal V^*)$ with image $W'\cong\mathbb P(\mathcal V^*)$, and we have a projective birational morphism $\nu':M'\to \overline M$ which is an isomorphism away from $W'$ and restricts to $\mathbb P(\mathcal V^*)\to Y$ on $W'$. The birational morphism $M\dashrightarrow M'$ is the **Mukai flop** of $M$ along $W$.

My question is as follows. Suppose we had all the hypotheses in the first paragraph above, and suppose we are *given* a projective birational map $\nu':M'\to \overline M$ which is an isomorphism away from $Y$ and which restricts to the dual projective bundle $\mathbb P(\mathcal V^*)\to Y$ away from $Y$. Can we then conclude that $M\dashrightarrow M'$ is the Mukai flop along $W$? If not, are there some additional conditions that we can impose on $\nu'$ which make it true?

Thanks kindly.