The nonvanishing characteristic-class conjecture for skew-framed immersions

Let $b=b(k)$ be the number of ones in the binary expansion of $k$, and assume $n+2\geq 2^{b(k)}$. Let $s=n-2b(k)+k+3$. Consider a skew-framed immersion

$$\alpha\in Imm^{sf,\kappa_1,\dots,\kappa_j}(b(k)-1,n-b(k)+1)$$

of a manifold $M^{b(k)-1}$ with normal data $\Xi_M$, characteristic classes $\kappa_M\in H^1(M^{b(k)-1};{\mathbb Z}/2)$ and $\kappa_i\in H^1(M^{b(k)-1};{\mathbb Z}/2)$ for $i=1,\dots,j$, such that $M^{b(k)-1}$ immerses in ${\mathbb R}^{2b(k)-2-k}$. The nonvanishing characteristic-class conjecture. There exists a collection $\{\xi_1,\dots,\xi_s\}$ of independent characteristic numbers of $\nu(M)=(n-b(k)+1)\kappa_M$ that does not vanish. The claim concerns the existence of a skew-framed immersion with prescribed normal data and nonzero characteristic numbers, relevant to the geometric approach to stable homotopy groups of spheres and the Kervaire invariant problem. The supplied text does not establish the conjecture or provide evidence resolving it.