Are intersections of essential sublocales essential?

Question

A sublocale $X_j$ of a locale $X$, given by a nucleus $j : \mathcal{O}(X) \to \mathcal{O}(X)$, is called essential (sometimes also principal) if and only if the following equivalent conditions are satisfied:

1. There is a monotone map $b : \mathcal{O}(X) \to \mathcal{O}(X)$ which is left adjoint to $j$.
2. The nucleus $j$ preserves arbitrary (not only finite) meets.
3. For any $u \in \mathcal{O}(X)$, there is a smallest $v \in \mathcal{O}(X)$ such that $u \preceq j(v)$.
4. The sheafification functor $\mathrm{Sh}(X) \to \mathrm{Sh}(X_j)$ possesses a left adjoint (it always has a right adjoint).
5. The geometric embedding $\mathrm{Sh}(X_j) \hookrightarrow \mathrm{Sh}(X)$ is an essential geometric morphism.

Question: Are arbitrary intersections of essential sublocales, taken in the partially ordered set of sublocales, again essential?

Remark: At least finite intersections of open sublocales (which are automatically essential) are again open (and therefore essential). The question is probably answered in the article On the complete lattice of essential localizations by Kelly and Lawvere, but unfortunately I wasn't able to obtain it.

Motivation: For essential sublocales $i : X_j \hookrightarrow X$, there is a simple description of the pullback functor $i^{-1} : \mathrm{Sh}(X) \to \mathrm{Sh}(X_j)$: $(i^{-1} \mathcal{E})(u) = \mathcal{E}(b(u))$ (no colimits or sheafification required). Since many interesting sublocales arise as intersections, it's therefore of interest to know whether essentiality is preserved by intersections.

Answer 1

The answer is most likely negative.

Arbitrary unions of essential sublocales are essential (and this is easy to show, using the explicit description of the nucleus associated to the union), but intersections of essential sublocales are probably not.

There are counterexamples in the cited article by Kelly and Lawvere, however they are in the slightly more general context of essential localizations of categories. As the title of their paper would suggest, they also show that the lattice of essential localizations is complete – but infima are not calculated as in the lattice of all localizations.

A few details are now written up at the nLab entry.