Assume that SAT ∈ PSIZE, does it imply that NP = coNP?
If $SAT \in PSIZE$ then the polynomial hierarchy collapses to the second level:
$\Sigma_2 = \Pi_2$ (see Karp-Lipton theorem ); but $NP=coNP$ (i.e. $\Sigma_1 = \Pi_1$) is stronger (the PH collapses to the first level).