Let $K$ be a non-perfect field of characteristic $2$. Let $T \subseteq K$ be a discrete valuation ring.
Assume there exist $a,b \in K^{\times}$ such that the projective conic $C$ defined by $$aX^2 + b Y^2 + Z^2=0$$ contains no $K$-rational point
(in particular $a,b$ and $ab$ are not squares in $K$). Then $C$ is non-smooth over $K$ at every point. However, $C$ is regular (see Exercise 4.3.22 (d) of Qing Liu's book *Algebraic Geometry and Arithmetic Curves*).

Question:Does $C$ have a regular projective model over $T$ (i.e. a regular fibered projective surface over $T$ with generic fiber isomorphic to $C$) ?

Every *smooth* curve over $K$ does admit a regular projective model over $T$, as is shown (for example) in Corollary 8.3.51 of Liu's aforementioned book.
This seems to be based on Lipman's resolution of singularities of $2$-dimensional *excellent* Noetherian schemes, applied to a normal model of the curve over $T$, or rather some base change to the completion of $T$ (excuse the sloppy description, this may not be a very accurate description of what is really going on).
In any case, it seems to be crucially used that the generic fiber of the normal model is not only regular, but smooth over $K$.

Nevertheless, this is a general statement, and it is a priori possible that in concrete cases of regular projective curves (such as all integral regular conics over $K$, which I am interested in) there do exist regular projective models over $T$, regardless of whether the regular curve is smooth over $K$ or not.