Kye's decomposability conjecture for 2-positive maps on $M_3(\mathbb{C})$

Let $M_n(\mathbb{C})$ be the algebra of all $n\times n$ complex matrices, and let $B(M_m(\mathbb{C}),M_n(\mathbb{C}))$ denote the space of linear maps from $M_m(\mathbb{C})$ to $M_n(\mathbb{C})$. A map is 2-positive if $id_2\otimes\phi$ is positive, and 2-copositive if $\tau_2\otimes\phi$ is positive, where $id_2$ is the identity map and $\tau_2$ is the transpose map. A positive map is decomposable if it is the sum of a completely positive map and a completely copositive map.

Kye's conjecture. Every 2-positive, respectively 2-copositive, linear map in $B(M_3(\mathbb{C}),M_3(\mathbb{C}))$ is decomposable.

The conjecture asks whether the decomposability property known for generalized Choi maps extends to every 2-positive or 2-copositive map on $M_3(\mathbb{C})$. The paper states that it is answered affirmatively by the subsequent theorem.