$L^p$-theory of forward-backward stochastic differential equations

For forward-backward stochastic differential equations (FBSDEs, for short), under certain conditions, one has the existence and uniqueness of an adapted $L^2$-solution. A natural question is whether such a uniquely existed adapted $L^2$-solution is actually an adapted $L^p$-solution for some $p \gt 2$, under proper conditions? Such a result has its own interest in the theory of FBSDEs and it also has some important applications in optimal stochastic control theory of FBSDEs. This paper addresses such an issue in certain extent and poses some open questions.