Does capillary action affect the accuracy of mercury barometer?

We know that mercury barometers are used to measure the atmospheric pressure by determining the height of mercury in the vertical column. Further, we know that the level of mercury in a capillary tube falls due to capillary action. The distance by which the level falls (due to capillary action) is given by Jurin's law:

$$h=\frac{2S\cos\theta}{r\rho g}$$

where $$h$$ is the rise or fall in height accordingly as it's positive or negative, $$S$$ is the surface tension, $$\theta$$ is the contact angle of the liquid on the tube wall, $$r$$ is the radius of the capillary tube, $$\rho$$ is the mass density and $$g$$ is the local acceleration due to gravity.

Almost in all mercury barometers apparatus, I noticed that the vertical column is not very large. It is thin or in other words its cross-sectional radius is small. From the Jurin's law we can say that the height is inversely proportional to the radius. So, I think, as the radius of the mercury column in the barometer is decreased the fall in the level due to capillary action increases. I believe this conclusion is correct.

If this is the case, I think the fall in mercury level will be misunderstood as fall in pressure. While learning about mercury barometers, I didn't see any correction terms for the fall in level due to capillary action, only the pressure due to mercury and the atmosphere was taken into account. So, does the capillary action affect the accuracy of a mercury barometer?

• @user47014: "capillary action changes with the mercury level" - This is not what I meant. I am saying capillary action causes decrease in the mercury level. Regarding your second statement - I don't think we take the fall in level due to capillary action into account. If it's taken into account then the concept of mercury barometers should have been introduced after describing capillary action in textbooks. Even in many sources on the internet, no one seems to include a correction factor for the same. Dec 1 '19 at 8:28
• Mercury barometers and manometers Pages 34-36 and 58. Dec 1 '19 at 10:44
• Not much because the radius the tube of barometer is quite high Dec 3 '19 at 0:18