Subdivided graphs preserve interval colorability

Let $G$ be a simple graph. Write $S(G)$ for the graph obtained by subdividing every edge of $G$ once, and let $\mathfrak{N}$ denote the class of graphs admitting an interval coloring.

Subdivision conjecture. If $G\in \mathfrak{N}$, then

$$S(G)\in \mathfrak{N}.$$

This would generalize the known result that the subdivision of every regular graph belongs to $\mathfrak{N}$; the statement for general simple graphs is posed as an interesting open problem.