A nice recent paper of Bezhanishvili and Kornell (which actually references this MathOverflow question specifically) has shown that the existence of products fails very badly in $\mathrm{Top}_{\mathrm{open}}$. In particular, the product of Sierpiński space $\mathbb{S}$ (i.e., the set $2 = \{0,1\}$ with the topology $3 = \{0,1,2\} = \{\varnothing, \{0\},\{0,1\}\}$) with itself does not exist in $\mathrm{Top}_{\mathrm{open}}$ (or in the full subcategories of $T_0$ and of sober spaces with open maps). Interestingly enough, they do this by mimicking the construction of the von Neumann hierarchy inside the product $\mathbb{S} \times \mathbb{S}$, showing that no small space satisfies the required universal property. As a consequence they show that the coproduct of the three-element Heyting algebra (i.e., the frame of opens of $\mathbb{S}$) with itself does not exist in the category of complete Heyting algebras. I think this resolves Todd's conjecture negatively, although I might be getting confused by the completeness/cocompleteness terminology.
This would also seem to contradict what you mention about the WordPress post, since the counterexample is a square, but I'm not entirely sure since the post has vanished from the internet.