Haefliger-type conjecture for coupled embeddings

Let $M$ and $N$ be smooth closed manifolds of dimensions $p$ and $q$, respectively. Write

$$\Delta=(\Delta M\times N^2)\cup(M^2\times\Delta N)\subset M\times N\times M\times N.$$

A $(\mathbb{Z}/2)^2$-equivariant map is understood with respect to the involutions exchanging the two copies of $M$ and the two copies of $N$. Haefliger-type conjecture for coupled embeddings. If

$$d>\frac{3}{2}(p+q)+\frac{3}{2},$$

then the existence of a differentiable coupled embedding $M\times N\to\mathbb{R}^d$ is equivalent to the existence of a $(\mathbb{Z}/2)^2$-equivariant map

$$g\colon M\times N\times M\times N\to\mathbb{R}^d$$

such that $g^{-1}(0)=\Delta$. This is proposed as the coupled-embedding analogue of Haefliger's theorem, which gives the corresponding equivalence for embeddings of a single smooth closed manifold; whether the stated equivalence holds in this range is left open.