I have a general question surrounding certain harmonic functions.

I was able to solve the Laplace equation $\Delta f = 0$ in $\mathbb{R^3}$, subject to two spherical (equal radii) boundary conditions, where the function takes on certain scalar values on both surfaces, and it goes to zero in the far field.

My solution is very messy, but there's a certain property that I need but it's difficult to prove. That is, assuming the function depends on spatial coordinates $x,y,z$ as well as distance $d$ between centres of the spheres, will it always be true that the size of the solution $f$ at the midpoint between the two spheres will always decrease monotonically as $d$ gets larger?

To be specific, assume the spheres are aligned on the $x$-axis with the origin being the midpoint and distance $d$ between their centres.

Why is it true that $|f(0,0,0)|$ decreases monotonically as $d$ gets larger?

Physically it makes sense because as $d$ gets larger, the balls should be further away. I have numerical reasoning to believe it's correct too.

I just am not sure how to prove it. Is there an argument involving maximum principle, or properties of harmonic functions anyone knows of? Has this problem already been solved that someone can provide a reference to? I'd appreciate it.

Note that I am also, in general, interested if this question can be extended for any pair of symmetric boundaries. I would propose that it should, and in this case I have considered spheres, though I am not sure how to show this.

Note: I had cross-posted this here on MSE, though since it's a research question I think it's best asked on this site.