Singularity conjecture for higher-degree spectrahedral boundaries

Let $A,B,C \in \operatorname{Sym}_r(\mathbb{R})$ be real symmetric matrices of size $r$, and set

$$f=\det(I+Ax+By+Cz).$$

Assume that $f$ is a polynomial of degree at least three. Singularity conjecture. The surface in complex projective three-space with affine equation $f=0$ is singular. This conjecture is suggested by the preceding result for cubic polynomials and by examples, but no characterization of spaces of symmetric matrices of rank greater than three is known; the general higher-degree case remains open.