[A nice recent paper of Bezhanishvili and Kornell](https://arxiv.org/abs/2407.13951v1 "The category of topological spaces and open maps does not have products") (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.