Let ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$ be a quaternion division algebra over $F$, endowed with its canonical involution. We denote by $F_Q$ the function field of the associated conic, which is the Severi–Brauer variety of $Q$. Since ${\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}}$ is of symplectic type, it becomes hyperbolic over any field that splits $Q$, hence in particular over $F_Q$. From this, one may easily deduce that *any $(A,\sigma)$ that contains ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$ becomes hyperbolic over $F_Q$.* The main theme of this paper is to investigate the reverse implication. In the case where $A$ is split and $\sigma$ is anisotropic, it is an easy consequence of the Cassels–Pfister subform theorem in the algebraic theory of quadratic forms that the converse holds, see Proposition \[split.prop\]. When $A$ is not split, the problem is much more delicate, and comparable to the characterization of quadratic forms that become *isotropic* over $F_Q$, which was studied by Hoffmann, Lewis, and Van Geel [@Hoff:thesis], [@HLVG], [@HVG]. Using an example from [@HVG] of a $7$-dimensional quadratic form over a suitable field $F$ that becomes isotropic over $F_Q$, we construct in §\[totdec.sec\] a division algebra $A$ of degree $8$ with an orthogonal involution $\sigma$ such that $(A,\sigma)$ becomes hyperbolic over $F_Q$ but does not contain ${(Q,{\overline{\rule{2.5mm}{0mm}\rule{0mm}{4pt}}})}$, even 