Why is the centripetal force non-conservative while the centrifugal isn't?

Why is the centripetal force non conservative while the centrifugal isn't?

As regards to the title of your post, the centripetal or centrifugal force may be conservative or non conservative. For example, if the centripetal force is gravity then it is conservative. The equal but opposite centrifugal force (pseudo force) that exists only in the rotating (non inertial) frame would then also be conservative. But if the force responsible for the circular motion is static friction, like that between the tires and road of a car enabling it to turn in a circle, then both the centripetal and centrifugal forces would be non-conservative. What makes a force conservative or non conservative is whether or not it can be written as the gradient of a potential.

Aren't both perpendicular to $R$? Also, doing $nabla times vec{F}$ using $hat{r}$, $hat{t}$, $hat{k}$ as versors (with $t$ tangent to $r$ and $k$ perpendicular to the surface) i get $0$ with both. Those informations are colliding.

If by $R$ you mean the radius of a circular path, the centripetal force acts along the radius pointing towards the center of the circle. Centripetal means "center seeking". In an inertial frame it is the only force acting on the object causing it to follow a circular path. If its removed the object would fly off moving in a straight line tangent to the path (tangent to $R$) per Newton's first law. In the non inertial (accelerating or rotating) frame the object would appear to fly off radially outward. The apparent force that is doing this, the centrifugal force, thus acts radially outward.

Regarding the rest, you need to further define the vectors in order to be able to comment.

Hope this helps.