Semistability conjecture for nonsplit tangent sheaves of codimension-one foliations on projective three-space

Let ${\mathscr{F}}$ be a codimension one foliation on ${\mathbb{P}}^{3}$, with tangent sheaf $T_{\mathscr{F}}$. A coherent sheaf is split if it is a direct sum of line bundles. Semistability conjecture. If $T_{\mathscr{F}}$ is not split, then it is $\mu$-semistable. The conjecture is motivated by results for rational foliations and logarithmic foliations of degree at most $2$, but it is false when one allows non-integrable codimension-one distributions.