Skein reduction and stabilization for the odd Khovanov spectra of $T(2,m)\sqcup T(2,n)$

For split links the even Khovanov spectrum is a smash product of the spectra of the pieces; the odd one is not. For $T(2,m)\sqcup T(2,n)$ we show that the odd spectrum is, in each quantum grading, a wedge of spheres and copies of a suspension of $\mathbb{CP}^{2}$. Using the skein cofiber sequence at