= Solution
On the affine chart $[1:z_1:\cdots:z_n]$ of $\mathbb{CP}^n$, the normalized <Fubini-Study form> is
$$
\omega_{\mathrm{FS}}
=\frac{i}{2}\,\partial\bar\partial
\log\!\left(1+\sum_{j=1}^n|z_j|^2\right).
$$
Its integral over a projective line is $\pi$. The <Darboux theorem (symplectic geometry)> says that every point of a symplectic $2n$-manifold has local coordinates in which the form is $\sum_jdx_j\wedge dy_j$.
To form the blowup $\widetilde X_\lambda$, choose a <symplectic embedding> of the closed standard ball $B^{2n}(\lambda)$ centered at $p$, remove its interior, and collapse each characteristic Hopf circle of its boundary $S^{2n-1}$ to a point. The boundary becomes the exceptional divisor $E\cong\mathbb{CP}^{n-1}$, and the reduced form extends the old form outside the ball with every projective line in $E$ having area $\pi\lambda^2$. This is the <symplectic blowup> of size $\lambda$. In real dimension four its volume is
$$
\operatorname{Vol}(\widetilde X_\lambda)
=\operatorname{Vol}(X)-\frac{\pi^2\lambda^4}{2}.
$$
Thus, whenever two different sizes are allowed, different $\lambda$ give different total symplectic volumes and hence nonsymplectomorphic blowups.
The punctured area-$\pi$ sphere $\mathbb{CP}^1\setminus\{\infty\}$ is symplectomorphic to the open unit disc. Consequently
$$
B^4(r)\subset D^2(1)\times D^2(1)
\hookrightarrow\mathbb{CP}^1\times\mathbb{CP}^1
$$
for every $r<1$. The bound is sharp. Given a hypothetical larger ball, choose a compatible almost complex structure agreeing with the pushed-forward standard structure on the ball. A standard J-holomorphic-curve result supplies a sphere in one ruling class through the ball center. Its total area is $\pi$, while monotonicity inside the ball requires at least $\pi r^2$. Hence $r\leq1$. Equivalently, the <Gromov width of the monotone product of projective lines> is
$$
\boxed{c_G(\mathbb{CP}^1\times\mathbb{CP}^1)=\pi.}
$$
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