Approximation conjecture for algebraic unoriented bordism classes

Let $V$ be a nonsingular real algebraic variety and let $M$ be a compact smooth submanifold of $V$. Suppose that the unoriented bordism class of the inclusion map $M\hookrightarrow V$ is algebraic. The Approximation conjecture. Then $M$ is $\varepsilon$-isotopic to a nonsingular Zariski locally closed subvariety of $V$. This conjecture seeks to approximate smooth submanifolds whose bordism classes are algebraic by nonsingular algebraic subvarieties. A slightly weaker assertion, with $M\times\{0\}\subset V\times\mathbb{R}$, is known, while the stated approximation remains open.