Let $T\in\mathscr{B(\mathcal{H})}$ and $P,Q$ are orthogonal projections on $\mathcal{H}$ with $\Vert P-Q\Vert<1$. Show that $$\text{dist}\left(\overline{W(PT|_{\text{ran}P})},\overline{W(QT|_{\text{ran}Q})}\right)\leq\Vert T\Vert\Vert P-Q\Vert\left[1+\frac{2}{1-\Vert P-Q\Vert^2}\right]$$ where $W(T)=\{\langle Tx,x\rangle:x\in\mathcal{H} \text{ and } \Vert x\Vert=1\}$, the numerical range of $T$ and $\text{dist}(A,B)=\inf\{\vert a-b\vert: a\in A, b\in B\}$, the distance between $A$ and $B$ for $A,B\subseteq\mathbb{C}$.
Attempt: I was trying to obtain the above bound using $\overline{W(T)}=\bigcap\limits_{\theta\in[0,2\pi)}\{\xi\in\mathbb{C}:\Re{(e^{i\theta}\xi)\leq\lambda_1(\Re{(e^{i\theta}T}))}\}$ but not able to establish. Any comment is highly appreciated. Thanks in advance.