I think the question is confusing. "Sincere" seems to imply free will of some form. The offer is certainly free - that is, there is nothing we offer in exchange for the promise of the Gospel. But we can not believe the Gospel unless we are called, given the faith to believe the Gospel.
If the "reprobate" are the un-elect, and "sincere" means "free to accept or reject"; then no, there is no sincere offer of the Gospel to the reprobate. If "sincere offer of the Gospel" means "everlasting life is offered freely to those who are given faith to believe", then yes, the offer is sincere.
But to even ask the question, is God is sincere, seems to question God's motives, or to imply man has free will. We are in no place to question God's motives, and human free will undercuts God's sovereignty.
If the "reprobate" are the un-elect, and "sincere" means "free to accept or reject"; then no, there is no sincere offer of the Gospel to the reprobate. If "sincere offer of the Gospel" means "everlasting life is offered freely to those who are given faith to believe", then yes, the offer is sincere.
But to even ask the question, is God is sincere, seems to question God's motives, or to imply man has free will. We are in no place to question God's motives, and human free will undercuts God's sovereignty.