= Solution
The oxygen-pressure exponents can be explained by a change in the dominant charge state of metal vacancies, with holes carrying the p-type current. For a divalent metal oxide, a doubly negative metal vacancy is $V_{\rm M}^{\prime\prime}$ in <Kröger–Vink notation>. A conventional oxygen-incorporation reaction is
$$
\frac12\mathrm O_2(g)\rightleftharpoons
O_{\rm O}^{\times}+V_{\rm M}^{\prime\prime}+2h^\bullet.
$$
Treating regular oxygen-site activity as constant, the <law of mass action> gives $K_2=[V_{\rm M}^{\prime\prime}]p_h^2/p_{\rm O_2}^{1/2}$. If these are the compensating defects, <charge neutrality> gives $p_h=2[V_{\rm M}^{\prime\prime}]$, hence $p_h^3\propto p_{\rm O_2}^{1/2}$ and \b[$\sigma\propto p_{\rm O_2}^{1/6}$], assuming nearly constant hole mobility.
For singly negative vacancies, use
$$
\frac12\mathrm O_2(g)\rightleftharpoons
O_{\rm O}^{\times}+V_{\rm M}^{\prime}+h^\bullet.
$$
Now $p_h=[V_{\rm M}^{\prime}]$ and $p_h^2\propto p_{\rm O_2}^{1/2}$, so \b[$\sigma\propto p_{\rm O_2}^{1/4}$]. The charge-state equilibrium $V_{\rm M}^{\prime}\rightleftharpoons V_{\rm M}^{\prime\prime}+h^\bullet$ gives $[V_{\rm M}^{\prime\prime}]/[V_{\rm M}^{\prime}]=K_{\rm ion}/p_h$: the singly charged fraction can therefore dominate as oxygen pressure and hole <concentration> increase. This is a consistent dilute-defect model, not a unique experimental identification of every defect. Changes in mobility, defect association or other compensating defects require independent tests.
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