Moore's extrinsic product conjecture for Riemannian products

Let $M^{n_1}_1$ and $M^{n_2}_2$ be Riemannian manifolds. Suppose that $M^{n_1}_1$ can be locally isometrically immersed in $\mathbb{R}^{n_1+p_1}$ but in no lower-dimensional Euclidean space, and that $M^{n_2}_2$ can be locally isometrically immersed in $\mathbb{R}^{n_2+p_2}$ but in no lower-dimensional Euclidean space. Moore's conjecture. Every isometric immersion

$$f\colon M^{n_1}_1\times M^{n_2}_2 \rightarrow \mathbb{R}^{n_1+n_2+p_1+p_2}$$

is an extrinsic product. The conjecture concerns when an isometric immersion of a Riemannian product must split into the corresponding factor immersions; the source presents it as a main inspiration and does not give evidence of a resolution.