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We study the capillary rise of viscous liquids into sharp corners formed by two surfaces whose geometry is described by power laws, $$h_i(x) = c_i x^n$$, $i = 1,2$, where $$c_2 > c_1$$ for $$n \geq 1$$. Prior investigations of capillary rise in sharp corners have shown that the meniscus altitude increases with time as $$t^{1/3}$$, a result which is universal, i.e., applies to all corner geometries. The universality of the phenomenon of capillary rise in sharp corners is revisited in this work through the analysis of a partial differential equation for the evolution of a liquid column rising into power-law-shaped corners, which is derived using lubrication theory. Despite the lack of geometric similarity of the liquid column cross-section for $n>1$, there exists a scaling and a similarity transformation that are independent of $$c_i$$ and $$n$$, which gives rise to the universal $$t^{1/3}$$ power-law for capillary rise. However, the prefactor, which corresponds to the tip altitude of the self-similar solution, is a function of $$n$$, and it is shown to be bounded and monotonically decreasing as $$n\to \infty$$. Accordingly, the profile of the interface radius as a function of altitude is also independent of $$c_i$$ and exhibits slight variations with $$n$$. Theoretical results are compared against experimental measurements of the time evolution of the tip altitude and of profiles of the interface radius as a function of altitude.