The minimum-edge non-TRVG conjecture for $K_{3,5}$

Let a transparent rectangle visibility graph (TRVG) be a graph admitting a representation by axis-parallel rectangles in which two vertices are adjacent when their rectangles are visible to one another, and call a graph that has no such representation a non-TRVG. The complete bipartite graph $K_{3,5}$ has $15$ edges.

Minimum-edge non-TRVG conjecture. $K_{3,5}$ is a non-TRVG with the least number of edges.

The paper establishes that $K_{3,5}$ is a non-TRVG and reports that no non-TRVG with fewer than $15$ edges was found; the conjectured minimality remains open.