Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram

$\require{AMScd}$ \begin{CD} 0 @>>> X_1 @>f_1>> X_2 @>f_2>> X_3 @>>> 0\\ @V VV @V T_1 VV @V T_2 VV @V T_3 VV @V VV \\ 0 @>>> Y_1 @>>g_1> Y_2 @>>g_2> Y_3 @>>> 0 \end{CD}

with two horizontal topologically short exact sequences. If $T_1$ and $T_3$ are nuclear operators, does it imply that $T_2$ is nuclear as well? A reference to problems of this general form, would be most welcome.