The upper-bound conjecture for critical points of real line arrangements

Let $F(x,y)$ be a polynomial whose zero set is a plane curve consisting of the product of $d$ real lines. Consider the critical points of $F$ having the same non-zero critical value. Upper-bound conjecture. The maximum number of such critical points is bounded by approximately $\frac{1}{3}d^2$. This concerns the extremal number of critical points relevant to constructing surfaces with many real nodes; the supplied text does not state whether the bound has been proved or remains open.