Nonexistence conjecture for an unbiased photon transfer conversion gain estimator

Let $Y_1$ and $Y_2$ be independent gamma random variables with densities

$$f_{Y_i}(y_i)=\frac{\beta_i^{\alpha_i}}{\Gamma(\alpha_i)}y_i^{\alpha_i-1}e^{-\beta_i y_i},\quad i=1,2,$$

and let $\kappa_i=\alpha_i/\beta_i$. An estimator $\mathscr T(Y_1,Y_2)$ is required to satisfy

$$\int_{\mathbb R^+\times\mathbb R^+}\mathscr T(y_1,y_2)f_{Y_1}(y_1)f_{Y_2}(y_2)\,\mathrm d(y_1,y_2)=\frac{1}{\kappa_1-\kappa_2}.$$

Nonexistence conjecture. There is no estimator $\mathscr T$ satisfying this equation when both $\kappa_1$ and $\kappa_2$ are unknown.

The claim concerns whether an unbiased estimator exists for the photon transfer conversion gain estimand $\tau=(\kappa_1-\kappa_2)^{-1}$. The preceding discussion establishes that inversion with respect to one parameter is possible when $\kappa_1$ is known, but describes the remaining inversion as intractable; no resolution of the unknown-parameter case is supplied here.