Strong Gross–Zagier conjecture for Heegner points

Let $E$ be an elliptic curve over $\mathbf{Q}$, let $K$ be an imaginary quadratic field satisfying the Heegner hypothesis, let $P_K\in E(K)$ be the Heegner point, let $C$ be the product of the local Tamagawa numbers at primes dividing the conductor, let $M$ be the Manin constant, let $u_K$ satisfy $2u_K=\#\mu(K)$, and let $\Sh(E/K)$ be the Tate–Shafarevich group. Strong Gross–Zagier conjecture. If $P_K$ has infinite order in $E(K)$, then $\mathbf{Z}P_K$ has finite index in $E(K)$ and

$$[E(K):\mathbf{Z}P_K]=u_K\cdot C\cdot M\cdot\left(\#\Sh(E/K)\right)^{1/2}.$$

This is obtained by equating the Gross–Zagier formula with the conjectural Birch–Swinnerton-Dyer formula; the supplied text gives no resolution status.