Can Anyone prove the following conjecture?

Consider $k$ rational function vectors $V_1(x_1,\cdots,x_n),\cdots,V_k(x_1,\cdots,x_n)$. They are called \textbf{linearly dependent} if there exists rational functions $\alpha_1(x_1,\cdots,x_n),\cdots,\alpha_k(x_1,\cdots,x_n)$ which are not identically zero such that \begin{align} \alpha_1(x_1,\cdots,x_n)V_1(x_1,\cdots,x_n)+\cdots+\alpha_k(x_1,\cdots,x_n)V_1(x_1,\cdots,x_n)=0 \end{align} This defines the rank of a matrix.

Conjecture: (Rank-1 Decomposition Conjecture) Let a $l \times m$ polynomial matrix \begin{align} A(x_1,\cdots,x_n)= \left( \begin{array}{ccc} a_{11}(x_1,\cdots,x_n) & \cdots & a_{1m}(x_1,\cdots,x_n) \newline \vdots & \ddots & \vdots \newline a_{l1}(x_1,\cdots,x_n) & \cdots & a_{lm}(x_1,\cdots,x_n) \end{array} \right) \end{align} is rank $k$ where $a_{ij}$ are linear functions whose coefficients are taken from $\mathbb{C}$. Then, there exists $l \times m$ polynomial matrices $A^{(1)}(x_1,\cdots,x_n),\cdots,A^{(k)}(x_1,\cdots,x_n)$ that satisfies the following three properties:

(i) $A(x_1,\cdots,x_n)=A^{(1)}(x_1,\cdots,x_n)+ \cdots + A^{(k)}(x_1,\cdots,x_n)$.

(ii) The rank of $A^{(i)}(x_1,\cdots,x_n)$ is $1$.

(iii) The elements of $A^{(i)}(x_1,\cdots,x_n)$ are linear functions on $x_1,\cdots,x_n$ with coefficients taken from $\mathbb{C}$.

Examples : (1) $\left( \begin{array}{ccc} x_1 & x_2 & x_3 \newline x_4 & x_5 & x_6 \end{array} \right)$

= $\left(\begin{array}{ccc} x_1 & x_2 & x_3 \newline 0 & 0 & 0 \end{array}\right)$ + $\left(\begin{array}{ccc} 0 & 0 & 0 \newline x_4 & x_5 & x_6 \end{array}\right) $

(2) $\left( \begin{array}{cccc} x_1 & x_2 & x_5 & x_5 \newline x_1 & x_2 & x_6 & x_6 \newline x_1 & x_2 & x_7 & x_7 \newline x_8 & x_9 & x_{10} & x_{11} \end{array}\right) $

$= \left( \begin{array}{cccc} x_1 & x_2 & 0 & 0 \newline x_1 & x_2 & 0 & 0 \newline x_1 & x_2 & 0 & 0 \newline 0 & 0 & 0 & 0 \\ \end{array} \right) + \left( \begin{array}{cccc} 0 & 0 & x_5 & x_5 \newline 0 & 0 & x_6 & x_6 \newline 0 & 0 & x_7 & x_7 \newline 0 & 0 & 0 & 0 \end{array} \right) + \left( \begin{array}{cccc} 0 & 0 & 0 & 0 \newline 0 & 0 & 0 & 0 \newline 0 & 0 & 0 & 0 \newline x_8 & x_9 & x_{10} & x_{11} \end{array} \right) $

(3) $\left( \begin{array}{cc} x_1 & x_4 \newline x_2 & x_5 \newline x_3 & x_6 \end{array}\right)$

$= \left( \begin{array}{cc} x_1 & 0 \newline x_2 & 0 \newline x_3 & 0 \end{array} \right) + \left( \begin{array}{cc} 0 & x_4 \newline 0 & x_5 \newline 0 & x_6 \end{array}\right) $

(4) $ \left( \begin{array}{ccc} x_1 & 2 x_1 & 3 x_1 \newline 2x_2 & 4 x_2 & 6 x_2 \newline x_3 & 2 x_3 & 4 x_3 \end{array} \right)$

$= \left( \begin{array}{ccc} x_1 & 2 x_1 & 3 x_1 \newline 2x_2 & 4 x_2 & 6 x_2 \newline x_3 & 2 x_3 & 3 x_3 \end{array} \right) + \left( \begin{array}{ccc} 0 & 0 & 0 \newline 0 & 0 & 0 \newline 0 & 0 & x_3 \end{array} \right) $