The disc embedding conjecture

Let $M$ be a $4$-manifold, let $f_i\colon (D^2,S^1) \looparrowright (M,\partial M)$ be a collection of immersed discs, and let $g_i\colon S^2 \looparrowright M$ be framed immersed spheres satisfying

$$\lambda(g_i,g_j)=0.$$

Assume that the $g_i$ are transverse spheres, so that

$$\lambda(f_i,g_j)=\delta_{ij}.$$

Disc embedding conjecture. The circles $f_i(S^1)$ bound disjointly embedded discs in $M$ with transverse spheres, inducing the same framing on $f_i(S^1)$ as the $f_i$.

The source states that the union of the topological surgery and $s$-cobordism conjectures is equivalent to this disc embedding conjecture. It is the key geometric input for the relevant four-dimensional surgery theory, and the source provides no evidence of resolution.