FX Forward pricing with correlation between FX and Zero-Cupon

I would like to extend my question about about FX Forward rates in stochastic interest rate setup: FX forward with stochastic interest rates pricing

We consider a FX process $X_t = X_0 \exp( \int_0^t(r^d_s-r^f_s)ds -\frac{\sigma^2}{2}t+ \sigma W_t^2)$ where $r^d$ and $r^f$ are stochastic processes not independent of the Brownian motion $W$. The domestic risk-neutral measure is denoted by $\mathbb Q^d$.

The domestic and foreign bank accounts are $\beta^d$ and $\beta^f$ respectively. The domestic and foreign zero-coupon bond prices of maturity $T$ at time $t$ are respective $B_d(t,T)$ and $B_f(t,T)$.

The domestic bond follows the SDE

$$\frac{dB_d(t,T)}{B_d(t,T)} = r^d_t \ dt + \sigma(t,T) \ dW^1_t$$

with deterministic initial conditions $B_d(0,T)$. The drift and volatility functions in the SDEs are all deterministic functions and $W^1$ and $W^2$ are standard Brownian motions such that $\langle W^1, W^2\rangle_t =\rho \ dt$.

Now consider domestic $\tau$-forward measure $\mathbb Q^{d,\tau}$

$$\left. \frac{d\mathbb Q^{d,\tau}}{d\mathbb Q^d}\right|_{\mathcal F_t} = \frac{B_d(t,\tau)}{\beta_t^dB_d(0,\tau)} = \mathcal E_t \left( \int_0 ^. \sigma(s,\tau) \ dW^1_s\right)$$

and the $\mathbb Q^{\tau}$-Brownian motions

$$W^{1,\tau}_. := W^1_. + \int_0 ^. \sigma(s,\tau) \ ds$$

$$W^{2,\tau}_. := W^2_. + \rho \int_0 ^. \sigma(s,\tau) \ ds$$

Question

I would like to calculate the non-deliverable FX forward rate. Since the fixing date $t_f$ is such that $t_f< T$, where $T$ is the settlement date, it implies to pass by the calculation following expectation:

$$\mathbb E^{\mathbb Q^d} _t \left[ \exp(-\int_t^T r^d_s ~ds)\ X_{t_f}\right]$$

I can get to this point

$$\mathbb E^{\mathbb Q^d} _t \left[ \exp(-\int_t^T r^d_s ~ds)\ X_{t_f}\right]= B_d(t,t_f)\ \mathbb E^{\mathbb Q^{d,t_f}}_t\left[ B_b(t_f,T)X_{t_f}\right].$$

From that point I am struggling to get through all the calculations. Is there a smart way to compute the last expectation?