A projective plane consists of a set of points, $\mathcal{P},$ a set of lines $\mathcal{L}$ and a relation $\mathbf{I}$ between points and lines called incidence having the following relation, denoted $\left(\mathcal{P},\mathcal{L},\mathbf{I}\right)$: (1) Given any two distinct points, there is exactly one line incident with both of them. (2) Given any two distinct lines, there is exactly one point incident with both of them. (3) There are four points such that no line is incident with more than two of them. It can be shown that a projective plane, finite or otherwise has an equal number of points and lines. Thus, for every finite projective plane there is an integer $N≥2$ such that the plane has $N^{2}+N+1$ points and lines and $N + 1$ points on each line and through each point. The number $N$ is called the order of the projective plane.
It can be shown the number of $1-$dimensional and $2-$ dimensional subspaces of the vector space $\mathbf{F}^{3}_{q}$ is equal to $$ \frac{q^3-1}{q-1}=q^2+q+1. $$ If a $1-$dimensional subspace is contained in a $2-$ dimensional subspace we say they are incident. In particular we can construct an incident relation $\left( \mathcal{P},\mathcal{L},\mathbf{I}\right),$ where $\mathcal{P}$ and $\mathcal{L}$ are the $1$ and $2$ dimensional subspace of $\mathbf{F}^{3}_{q}$ respectively and the relation $\mathbf{I}$ is subspace containment. From this we see that a finite projective plane of order $q$ is a $2-\left(q^2+q+1,q+1,1\right)$ design. Usually this design is denoted by $PG\left(2,q\right).$ Following this vector space construction there exist a finite plane of order $N=q^{n}$ for each prime number $q.$
Indeed it appears the only finite projective planes that are known are of prime power order. And it has been conjectured that any finite projective plane must have prime power order. This conjecture is apparently still open. A great deal of work in studying the structure (existence, ovals etc.) of finite projective planes involves tools from the character and modular representation theory of finite groups. A major result in the field is the Bruck–Ryser–Chowla: If a finite projective plane of order $q$ exists and $q\equiv 1,2\pmod 4$, then $q$ must be the sum of two squares. This result for example can be used rule out a finite projective plane of order 6.
Question: What makes this conjecture difficult to solve?
Explicitly I am asking where are the stumbling blocks, no pun intended, within finite group and combinatorial theory that has made this problem impossible to conquer ? Incidentally, no pun I swear, there is an extensive body of literature devoted to computational aspects of finite projective planes. A great deal of work has been done establishing the (non-)existence of finite projective planes with or without prescribed properties. None of those efforts have resolved the conjecture.
Update:
This is a response to a series of comments written by @verret:
Following Brauer, Alberts, Hall ,Thompson and others a great body of research on finite projective planes has been approached ostensibly by the character theory and modular representation theory of finite groups. Since the time this conjecture was stated group theorist have: (1) classified all finite simple groups and (2) built a great machinery around the representation of finite groups. From the combinatorial side we have deep results from Combinatorial Commutative Algebra and the study of posets. Yet this conjecture remains unsolved. What is central to this conjecture that has escaped some of the best advancements of finite group theory and combinatorics ?