Skein reduction and stabilization for the odd Khovanov spectra of T(2,m) 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) T(2,n) we show that the odd spectrum is, in each quantum grading, a wedge of spheres and copies of a suspension of CP². 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 CP² summand of T(2,3) T(2,3), due to Schütz and Rajapakse, persists in quantum grading $m+n+8$ for all m,n≥3. We conjecture that the number of copies of CP² equals the number of copies of RP²² in the even spectrum.