Assume that SAT ∈ PSIZE, does it imply that NP = coNP?

Question

Assume that $\mathrm{SAT} \in \mathrm{PSIZE}$, does it imply that $\mathrm{NP} = \mathrm{coNP}$ ?

I think that I've managed to show that if $\mathrm{SAT} \in \mathrm{PSIZE}$, then both $\mathrm{NP}$ and $\mathrm{coNP}$ are contained in $\mathrm{PSIZE}$, but I can't see how does help me. Any ideas ?