Real critical points conjecture for reciprocal likelihoods of d-pencils

Let $\mathcal{L}$ be a d-pencil, meaning a pencil of quadrics with no zeros in real projective space, and suppose that it has $r$ distinct eigenvalues. For data $s=(s_1,\ldots,s_n)\in\mathbb{R}^n$, consider the reciprocal log-likelihood function

$$\widetilde{\ell}_S(x,y)=\sum_{i=1}^n\left(-\log(a_i x+y)-\frac{s_i}{a_i x+y}\right).$$

Real critical points conjecture. There exists $s=(s_1,\ldots,s_n)\in\mathbb{R}^n$ such that $\widetilde{\ell}_S$ has $2r-3$ distinct real critical points.