Maximum dimension of subspaces with no product basis

Let $n\ge2$ and $d_1,\ldots,d_n\ge2$ be integers, $\mathcal{F}$ a field. A vector $u\in\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ is called product if $u=u^{[1]}\otimes\cdots\otimes u^{[n]}$ for some $u^{[1]}\in\mathcal{F}^{d_1},\ldots,u^{[n]}\in\mathcal{F}^{d_n}$. basis composed of vectors basis. In this paper, we show that the maximum dimension subspaces $\mathcal{F}^{d_1}\otimes\cdots\otimes\mathcal{F}^{d_n}$ with no equal to $d_1d_2\cdots d_n-2$ either (i) $n=2$ or (ii) $n\ge3$ $\#\mathcal{F}>\max\{d_i : i\not=n_1,n_2\}$ $n_1$ $n_2$. When $\mathcal{F}=\mathbb{C}$, result related number simultaneously distinguishable states in general probabilistic theories (GPTs).