Suppose
\begin{equation}
M= \begin{bmatrix}
M_{11} & \cdots &M_{1d} \\
\vdots & \ddots & \vdots \\
M_{d1} & \cdots & M_{dd}
\end{bmatrix}
\end{equation}
is a $d \times d$ block matrix such that $M_{jk} = \sum_{i=1}^{r}(A_i)_{jk} B_i$, for some $A_i \in M_d(\mathbb{C})$, $B_i \in M_{n}(\mathbb{C})$ and $d,n,r >2$. Now, let
\begin{equation}
M^{\prime}= \begin{bmatrix}
M^T_{11} & \cdots &M^T_{1d} \\
\vdots & \ddots & \vdots \\
M^T_{d1} & \cdots & M^T_{dd}
\end{bmatrix},
\end{equation}
where $M^T_{jk} = \sum_{i=1}^{r}(A_i)_{jk} B^T_i$. Is it true that $\frac{R(M^{\prime})}{R(M)} \leq r$, where $R$ denotes the matrix rank? (for $r=1,2$ this statement is true!)