Codimension-1 Grassmannian embedding conjecture

Let $G_{k_i}(\boldsymbol{R}^{n_i})$ denote the Grassmannian of $k_i$-planes in $\boldsymbol{R}^{n_i}$, where $k_i\geq 1$ and $i\in\left\{1,2,3\right\}$. A smooth embedding of codimension $1$ is an embedding whose image has codimension $1$ in the target Grassmannian.

Codimension-1 Grassmannian embedding conjecture. The only codimension-$1$ smooth embedding

$$G_{k_1}(\boldsymbol{R}^{n_1})\times G_{k_2}(\boldsymbol{R}^{n_2}) \rightarrow G_{k_3}(\boldsymbol{R}^{n_3})$$

is the Segre embedding

$$\mathop{\mathbf{R}\mathrm{P}}\nolimits^1 \times \mathop{\mathbf{R}\mathrm{P}}\nolimits^1 \rightarrow \mathop{\mathbf{R}\mathrm{P}}\nolimits^3.$$

This conjecture extends the preceding observation from projective spaces to nonorientable real Grassmannians. It asserts that, apart from the Segre embedding of two projective lines into projective $3$-space, no product of three specified real Grassmannians admits a codimension-$1$ smooth embedding of the stated form.