Gromov's approximation and interpolation conjectures for elliptic manifolds

Let $Y$ be a connected complex manifold satisfying Condition $\mathrm{Ell}_1$. Let $X$ be a complex manifold, let $\Omega\subset X$ be a Runge domain, and let $f_0\in\mathcal{O}(X,Y)$ and $f_1\in\mathcal{O}(\Omega,Y)$ be holomorphic maps. Also let $X$ be a Stein manifold when considering the interpolation assertion, and let $D\subset X$ be a closed discrete subset with a map $\varphi:D\to Y$.

Gromov's conjectures. The following assertions hold:

1. If $f_1$ is homotopic to $f_0|_\Omega$ through holomorphic maps, then $f_1$ can be approximated on $\Omega$ by holomorphic maps $X\to Y$.
2. There exists $f_1\in\mathcal{O}(X,Y)$ homotopic to $f_0$ and satisfying $f_1|_D=\varphi$.

These are approximation and interpolation forms of the Oka principle for manifolds satisfying Condition $\mathrm{Ell}_1$; the source states that they receive affirmative answers from the paper's main theorem, so they are solved rather than open conjectures.