# What is expected (border) rank of the knonecker product of 3-tensors

Given two three order tensors $$T$$ and $$S$$ in $$F^{m\times n\times p}$$ and $$F^{a\times b\times c}$$. Clearly $$\operatorname{rk}(T\otimes S)\le \operatorname{rk}(T)\operatorname{rk}(S)$$. Does the equality holds for generic $$T$$ in $$F^{m\times n\times p}$$ and generic $$S$$ in $$F^{a\times b\times c}$$ (tensors are viewed as projective varieties via Segre embedding).

There are formats in which the equality is false for generic tensors: take $$F^{n \times n \times 1}$$ and $$F^{n \times n \times n^2}$$. Generic tensors in both formats are isomorphic to matrix multiplication tensors $$\left$$ and $$\left<1,n,n\right>$$ with ranks $$n$$ and $$n^2$$ respectively, but the rank of their tensor product $$\left$$ is less than $$n^3$$ for $$n \geq 2$$. If you don't like dimension $$1$$, the same happens for $$F^{n \times n^2 \times n}$$ and $$F^{n \times n \times n^2}$$.
For formats in which the subgeneric rank tensors form a hypersurface, the techniques of the paper "Border Rank Is Not Multiplicative under the Tensor Product" by Christandl, Gesmundo and Jensen, I believe, can be used to prove that for a generic tensor $$R(T^{\otimes 2}) < R(T)^2$$, but this does not extend to products of different tensors.
On the other hand, for generic $$2 \times 2 \times 2$$ tensors multiplicativity is true.