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 the level of spectra, we prove that for fixed $q-m-n$ the number of these copies is determined by finitely many small links and stabilizes for large $m$ and $n$. The $\mathbb{CP}^{2}$ summand of $T(2,3)\sqcup T(2,3)$, due to Schütz and Rajapakse, persists in quantum grading $m+n+8$ for all $m,n\ge3$. We conjecture that the number of copies of $\mathbb{CP}^{2}$ equals the number of copies of $\mathbb{RP}^{2}\wedge\mathbb{RP}^{2}$ in the even spectrum.