Duals of higher real $K$-theories at $p=2$

We study $\mathrm{K}(h)$-local Spanier-Whitehead duality for $C_{2^n}$-equivariant Lubin-Tate spectra, $E_h$, at the prime $2$ and heights $h$ divisible by $2^{n-1}$. We determine a $C_{2^n}$-equivariant equivalence $DE_h\simeq\Sigma^{-V_h} E_h$, for an explicit $C_{2^n}$-representation, $V_h$. We then study the $\mathrm{RO}(C_{2^n})$-periodicities of $E_h$ at some low heights. With these ingredients, we determine the self-duality of some higher real $K$-theories up to a specified suspension shift, at some low-heights. In particular, we show that $DE_4^{hC_8}\simeq \Sigma^{112}E_4^{hC_8}$.