Orbit D_3 stability conjecture under best rank-1 deflation

Let $\underline{\bf X}$ be a tensor in orbit $D_3$, and let $\underline{\bf Y}$ be a best rank-$1$ approximation of $\underline{\bf X}$. Orbit $D_3$ stability conjecture. Almost all tensors $\underline{\bf X}-\underline{\bf Y}$ are in $D_3$. Numerical experiments support this claim for generic tensors in orbit $D_3$, while explicit examples show that best rank-$1$ deflation can produce other orbits for tensors in the related orbits $G_2$ and $D_3$; the stated almost-everywhere assertion remains open in the source.