Oriented-boundary conjecture for algebraic models of 4k-manifolds

Let $M$ be a compact connected oriented smooth manifold of dimension $4k$. An algebraic model of $M$ is a nonsingular real algebraic variety diffeomorphic to $M$. Suppose there exists an algebraic model $X$ of $M$ such that every regular map

$$X\longrightarrow\mathbb{S}^{4k}$$

is null homotopic.

Oriented-boundary conjecture. The disjoint union of two copies of $M$ is an oriented boundary.

The preceding theorem gives this boundary condition as sufficient for constructing an algebraic model with the stated null-homotopy property when the dimension is divisible by four. The conjecture asserts that, in dimension $4k$, this condition is also necessary; the source provides no resolution status.