$G_\delta$-diagonal and productivity of the CCC

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.

• Isn't it enough to take the Moore Machine $M(L)$ over a Suslin Line $L$? Moore spaces have $G_\delta$ diagonals, the ccc is preserved by the Moore Machine construction and the square would be non-ccc because $M(L)$ maps continuously onto $L$ and $L^2$ is not ccc. – Santi Spadaro Dec 18 '17 at 22:28
• @SantiSpadaro What is the Moore machine construction? (Google is just turning up things in theoretical computer science, not topology.) – Noah Schweber Dec 18 '17 at 23:25
• @SantiSpadaro After some searching. Yes that is enough. Here is a paper by van Douwen and Reed were they establish the result. sciencedirect.com/science/article/pii/016686419190076X – Not Mike Dec 18 '17 at 23:28
• @NoahSchweber Based on what I'm finding, it should probably be called Reed's Machine. – Not Mike Dec 18 '17 at 23:37
• @NoahSchweber It's a machine that turns every first-countable regular space into a Moore space, and preserves many original properties of the original space, for example, chain-condition type properties are usually preserved. Correct, Not Mike, the Moore Machine is due to Mike Reed, but by Boyer's Law, mathematical discoveries are never named after their author. Not even Boyer's Law is due to Boyer, by the way. :-) – Santi Spadaro Dec 19 '17 at 11:31