On the nonexistence of a Green functor with values spin${}^c$ bordism and spin bordism

In this note, we show that there does not exist a $C_2$-ring spectrum whose underlying ring spectrum is ${\mathrm{M}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}}^c$ and whose $C_2$-fixed point spectrum is $\mathrm{M}\mathrm{S}\mathrm{p}\mathrm{i}\mathrm{n}$.