Does $f : p_\theta\mapsto p_{T\,\mid\,\theta}$ being injective imply statistic $T $ is sufficient?

Wikipedia says ... consider the map $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ which takes each distribution on model parameter $\theta$ to its induced distribution on statistic $𝑇$ . The statistic $T$ is said to be complete when $f$ is surjective, and sufficient when $f$ is injective . (emphasis mine) Is this claim true? ie does $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ being injective imply statistic $T$ is sufficient? My research so far : I think wikipedia is incorrect , as I can prove by counterexample. ie I can provide an example where $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ is injective but $T$ is not a sufficient statistic. Consider this conditional probability distribution of samples $X$ given parameters $\theta$ , ie $p_{X\,\mid \,\theta}$ : (table 1) $\theta_1$ $\theta_2 $ $x_1$ $0.1 $ $0.2 $ $x_2$ $ 0.2$ $0.2 $ $x_3$ $0.3$ $0.3 $ $x_4$ $ 0.4 $ $0.3 $ and here is the map of samples $ X$ to statistic $T$ , meaning that statistic $T$ calculated for sample $x_i$ (column 1) has value equal to $t_j$ (column 2) ie $T(x_i)=t_j$ : (table 2) sample statistic $x_1$ $t_1$ $x_2$ $t_1$ $x_3$ $t_2$ $x_4$ $t_2$ which leads to the following conditional probability distribution of statistic $T$ , given parameters $\theta$ , ie $p_{T\,\mid\,\theta}$ : (table 3) $\theta_1$ $\theta_2$ $t_1$ $0.3$ $0.4$ $t_2$ $0.7$ $0.6 $ In this case $f:p_{\theta }\mapsto p_{T\,\mid\, \theta }$ is injective (can be deduced from table 3), but the statistic $ T$ is not sufficient, as for a giv…