Bounds on the tensor and border rank ratios of tensor unfoldings?

Consider $T_1 \in \mathbb{C}^{d_1 \times \cdots \times d_k}$ as an order-$k$ tensor of the format $(d_1, \cdots , d_k)$. Now, let $T_2$ be an unfolding of $T_1$ that is obtained by merging $\mathbb{C}^{d_i}$ and $\mathbb{C}^{d_j}$ into a single vector space $\mathbb{C}^{d_id_j}$, i.e. $T_2$ is an order-$(k-1)$ tensor of the format $(d_1, \cdots , d_id_j, \cdots , d_k)$, for $i,j \in [k]$. The question is what are the best general lower and upper bounds on the ratio $\frac{R(T_1)}{R(T_2)}$, where $R(T_l)$ denotes the tensor rank of $T_l$, $l=1,2$? Same question can also be asked about $\frac{\underline{R}(T_1)}{\underline{R}(T_2)}$, where $\underline{R}$ denotes the border rank?

• I doubt if there are any results known at all for the bounds you are asking about. Possibly relevant are some bounds on flattening ranks due to Carlini and Kleppe, "Ranks derived from multilinear maps", JPAA (2011), mathscinet.ams.org/mathscinet-getitem?mr=2776439. – Zach Teitler Feb 14 '18 at 19:19
• Thank you @Zach! It seems that our knowledge of tensor and border ranks is not very advanced. Many relatively "basic" questions are still open! – SMD Feb 14 '18 at 19:49