Bilinear forms and the $\Ext^2$-problem in Banach spaces

Let $X$ be a Banach space and let $\kappa(X)$ denote the kernel of a quotient map $\ell_1(\Gamma)\to X$. We show that $\Ext^2(X,X^*)=0$ if and only if bilinear forms on $\kappa(X)$ extend to $\ell_1(\Gamma)$. From that we obtain i) If $\kappa(X)$ is a $\mathcal L_1$-space then $\Ext^2(X,X^*)=0$; ii) If $X$ is separable, $\kappa(X)$ is not an $\mathcal L_1$ space and $\Ext^2(X,X^*)=0$ then $\kappa(X)$ has an unconditional basis. This provides new insight into a question of Palamodov in the category of Banach spaces.