Fractional covering conjecture for separated d-intervals

Let $H$ be a hypergraph of separated $d$-intervals. Write $u(H)$ for its matching number and $\tau^*(H)$ for its fractional covering number. Fractional covering conjecture. One should have

$$\tau^*(H) \le d\nu(H).$$

This is known to fail for non-separated $2$-intervals, where an intersecting family can have $\tau^*\geq\tfrac{11}4$; whether every intersecting family of non-separated $2$-intervals satisfies $\tau^*<3$ remains open.