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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 that 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 exist 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.
