The converse between local compactness and local solidity of the unit filter

Let $X$ be a topological space. The set ${\left\{ 1 \right\}}$ is an element of the lattice of multiplicative filters $\operatorname{mF}(\mathcal{O}(X))$.

Local compactness–local solidity conjecture. $X$ is locally compact if and only if

$${\left\{ 1 \right\}} \in \operatorname{mF}(\mathcal{O}(X))$$

is locally solid.

The forward implication follows from the fact that a locally compact topological space induces a quantale whose unit filter is locally solid; the converse is posed as an open question.