Near-critical Ising interfaces in the thermal direction: Absolute continuity with respect to SLE$_3$

The scaling limits of interfaces in the critical two-dimensional Ising model are described by the Schramm--Loewner evolution $\mathrm{SLE}_3$. While, in the scaling limit, the law of the whole near-critical FK-Ising configuration is known to be singular with respect to the critical law in the therma