When solute dissolves in a solvent, is the resulting solution truly homogenous at infinite time when entropy is maximum? When a solute dissolves spontaneously in a solvent, the distribution of the solute tends to become homogenous.
For a truly homogenous mixture, there is uniform distribution of the solute in the solvent. But, only if the solute particles are equidistantly distributed, there will be perfect homogenity. That is, if there is no equidistant distribution, there will be certain "regions" where there is more solute concentration than the rest of the solution, hence no homogenity. But if there is equidistant distribution of the solute to attain homogenous, the system will be in a highly ordered state, implying the system has minimum entropy.
But, it is studied that, as solute dissolves the entropy maximises. 
So, does the final state at infinite time, of a system of solute and solvent, have homogenity or maximum entropy?
 A: 
But if there is equidistant distribution of the solute to attain homogenous, the system will be in a highly ordered state, implying the system has minimum entropy.

It is confusing how "ordered" is used when describing entropy. What makes something ordered? That could be a matter of opinion!
Imagine you have 3 coins. Each coin has 2 possible states: heads and tails. What are all the possible states if we flip all 3 coins?

H H H
H H T
H T T
T T H
T H H
H T H
T H T
T T T

There are 9 possible states of the system. Do you notice how only two of these states have all heads or all tails? The most likely states are mixed up! The states with all heads or all tails look pretty and more "ordered". 
In analogy with your situation, the all heads or all tails states would be ones where the solution decides to separate into its two components. This is extremely unlikely, but it is not impossible. 
So maximum entropy doesn't mean everything is mixed up perfectly. It does mean that when you check the system, you are far more likely to find things in a mixed up state than an "ordered" one.   
A: Your question is based on a chain of implications, most of them containing some quite common misunderstanding in statistical thermodynamics.

For a truly homogeneous mixture, there is uniform distribution of the solute in the solvent. But, only if the solute particles are equidistantly distributed, there will be perfect homogeneity.

True. In a homogeneous mixture, there is uniform distribution of the solute in the solvent. More formally, if we introduce the number density of the solute, $\rho_{solute}({\bf r})$, we have:
$$
\rho_{solute}({\bf r})=\mathrm {constant}.
$$
However, $\rho_{solute}({\bf r})$ is an average quantity, therefore its constant value implies that in every small volume element there will be in average the same number of solute particles. In turn, this implies the existence of an average inter-particle distance. It does not imply that particles are always equidistantly distributed.

That is, if there is no equidistant distribution, there will be certain "regions" where there is more solute concentration than the rest of the solution, hence no homogeneity. But if there is equidistant distribution of the solute to attain homogeneous, the system will be in a highly ordered state, 

As noted above, since homogeneity is equivalent to a constant average density, in a scale of ordering where the crystalline solid is the most ordered structure, corresponding to a non-constant $\rho_{solute}({\bf r})$ having the same spatial periodicity of the crystalline lattice, $\rho_{solute}({\bf r})=\mathrm {constant}$ represents the maximum spatial disorder. 
It may appear that some arbitrariness or subjective judgement is connected with this scale of ordering. It is true, and this is pointing to the inconsistency of using spatial disorder as a proxy of entropy, but for very special cases like weakly interacting systems.

implying the system has minimum entropy.

In any case, this last implication (maximum spatial order $\Rightarrow$ minimum entropy) is not always true. Otherwise, it would be impossible to have stable phase separated mixtures. Mixing seed oil and water in a glass allows everybody to check that phase separation may be the stable configuration as well as global homogeneity for other systems. What makes the difference is the presence of different inter-particle interactions. 
When connecting entropy with disorder, one should bear in mind that the only general connection between these two concepts, justified by statistical mechanics, is that disorder should be measured by the spread of probability of the set of states. It coincides with a uniform one-particle density only if interactions are small enough.
A: A homogeneous mixture is one in which each subsection of the mixture looks identical to every other. 
Consider the initial state where you had the solute and solvent well separated. You could easily point out where the solute is and where the solvent is. And each individual solute/solvent particle was moving around a smaller region of space. 
However, once they mix to form a homogeneous solution, the region of space each individual particle can move around is much larger. Here it’s not possible to point out where each particle will move in the future. 
The following image might help in visualising the above argument. 
Initially you can clearly demarcate red from blue and say here are my solute particles (red) and here are solvent particles. But at a later time, you can no longer make that distinction. In other words if someone asks what kind of particle is present at such and such position inside the box, you can easily say red or blue initially. But it gets more and more difficult to do so as time goes on. 
Obviously, if you sit and track each and every particle, you will be able to, but that is practically impossible (~$10^{23}$ particles). 
It is in this sense the homogeneous mixture is more “disordered” than the initial one. 
