Holomorphic factorization conjecture for null-homotopic symplectic mappings

Let $X$ be a finite dimensional reduced Stein space, and let $f\colon X\to \operatorname{Sp}_{2m}(\mathbb{C})$ be a holomorphic mapping that is null-homotopic. Let $M_1,\ldots,M_K$ denote the holomorphic matrix factors associated with vectors in $\mathbb{C}^{m(m+1)/2}$. Holomorphic factorization conjecture. There exist a natural number $K$ and holomorphic mappings

$$G_1,\ldots,G_K\colon X\to \mathbb{C}^{m(m+1)/2}$$

such that

$$f(x)=M_1(G_1(x))\dots M_K(G_K(x)).$$

The statement would extend the factorization result from $\operatorname{Sp}_4(\mathbb{C})$ to $\operatorname{Sp}_{2m}(\mathbb{C})$ whenever the relevant proposition holds for the successive values of $n$. Its status is not resolved in the supplied source context.