Is there a known example of a completely regular c.c.c. space with $G_\delta$-diagonal which is not productively c.c.c.?

The non-existence of such a space is consistent (for example, under $MA$ no such space should exist.) This would entail, I'm looking for a consistent example only; unless of course it's a theorem that completely regular c.c.c. spaces with $G_\delta$-diagonal are productivity c.c.c.

Edit: removed not very well thought out question.