Wang–Xu conjecture for very exotic spheres

Let a very exotic sphere mean a homotopy sphere that does not bound a parallelizable manifold. The preceding results show that very exotic spheres exist in at least $99$ congruence classes of dimensions modulo $192$. Wang–Xu conjecture. Very exotic spheres exist in all dimensions greater than $4$, except dimensions $5$, $6$, $11$, $12$, $27$, $43$, $56$, and $61$. This conjecture concerns the detection of elements in the cokernel of the stable $\mathrm{J}$-homomorphism and would determine precisely the dimensions in which very exotic spheres occur. The low-dimensional computations of Isaksen, Wang, and Xu, together with work of Ravenel and recent results, verify it up to dimension $102$; the assertion remains open in general.