Degree-parity conjecture for regular maps from products of spheres

Let $n$ and $m$ be positive integers. A continuous map is considered from the product of spheres $\mathbb{S}^n\times\mathbb{S}^m$ to $\mathbb{S}^{n+m}$, and $\binom{n+m}{n}$ denotes the corresponding binomial coefficient. Degree-parity conjecture. The following conditions are equivalent:

1. Every continuous map from $\mathbb{S}^n\times\mathbb{S}^m$ into $\mathbb{S}^{n+m}$ is homotopic to a regular one.
2. The number $\binom{n+m}{n}$ is odd.

This conjecture extends the equivalence established in the paper for $n\leq 3$ and predicts a complete parity criterion for algebraic approximation of maps from products of spheres. The paper provides new families where the criterion holds, but does not state a resolution in general.