This has since been explored and it appears as though there is an interesting type of moonshine for the Thompson group involving weakly holomorphic weight 1/2 modular forms over $\Gamma_0(N)$ with non-trivial multiplier systems.
To bring out the structure of this moonshine, we can consider the form
\begin{align}
\mathcal{F}_3(\tau) &\equiv 2b(\tau) + 240\theta(\tau) = \sum_{\substack{m\geq -3 \\ m\equiv 0,1~\mathrm{mod}~ 4}}c(m)q^m \\
&= 2q^{-3} + 248 + 2\cdot 27000q^4 - 2\cdot 85995q^5 + 2\cdot 1707264 q^8 \\
& \ \ \ \ \ \ \ \ \ \ \ \ \ - 2\cdot 4096000q^9 + 2\cdot44330496q^{12} + \cdots
\end{align}
where $$\theta(\tau) = \sum_{n\in \mathbb{Z}}q^{n^2}.$$
Inspection of the character table of Th suggests that we should interpret these first few coefficients as either single, real irreducible representations (with multiplicity 2) or representations of the form $V\oplus \overline{V}$. If we are right in making this association, it might further suggest the existence of a graded module $$W = \bigoplus_{\substack{m\geq -3 \\ m\equiv 0,1 ~\mathrm{mod}~ 4}}^{\infty}W_m$$ where we demand that the module be compatible with the Fourier coefficients of $\mathcal{F}_3$ in the sense that $\dim W_m = |c(m)|$. The alternating signs can be treated by endowing $W$ with a superspace structure. In other words, we can decompose each component into an odd and even part, $W_m = W_m^{(0)} \oplus W_m^{(1)}$. The multiple of $\theta$ in the definition of $\mathcal{F}_3$ was chosen so that each Fourier coefficient could be decomposed into dimensions of irreducibles of the same sign. Then, this super-module is special in that, for $m\geq 0$, $W_m$ has vanishing odd part when $m$ is even and vanishing even part when $m$ is odd. The coefficient of $q^{-3}$ in this sense has the 'wrong sign' in that $W_{-3}$ has vanishing odd part, a feature similar to one observed in Umbral moonshine.
One can take this idea farther and consider McKay-Thompson series as in Monstrous Moonshine, $$\mathcal{F}_{3,[g]}(\tau) = \sum \mathrm{str}_{W_m}(g)q^m$$ where the $\mathrm{str}$ is meant to denote the super trace. It was found that you can naturally identify these McKay-Thompson series with weight half modular forms (obtained via the formalism of Rademacher series) on $\Gamma_0(4\cdot o(g)\cdot m)$, i.e. of level 4 times a multiple of the order of $g$, with non-trivial multiplier system. It was verified that these candidates for the McKay-Thompson series produced a well-defined character for the components $W_m$ for $m\leq 52$. In other words, treating each MT series as a graded character, one can decompose the $W_m$ into irreducible representations of the Thompson group with positive integer multiplicities for $m\leq 52$ and it is conjecture that this holds more generally for all $m$.
As it turns out, this moonshine is also conjectured to enjoy a discriminant property similar to the one observed in Umbral moonshine, and also has a natural connection to Monstrous moonshine in that the Borcherds lift of $b(\tau)-4\theta(\tau)$ is the $T_{3C}$ McKay Thompson series from Monstrous moonshine, as was pointed out in earlier comments.
If you'd like to find out more about this moonshine, all this information was taken from the following paper, written by Jeff Harvey and me: http://arxiv.org/abs/1504.08179
Best Answer
It is not true anymore that a proof of this conjecture would lead to significant simplifications. Peterfalvi proved in 1984 a weaker version of this conjecture, which suffices to get rid of the chapter involving generators and relations in the original paper. Bender and Glauberman reproduce this argument in their book in one of the appendices, and it takes only two or three pages. Surprisingly, although the statement is about the solvability of a certain equation in finite fields, the proof is group theoretic and not number theoretic in nature. Moreover, this statement does not follow in an obvious way from standard results in number theory such as Weil's estimates. Anyway, it is highly questionable whether replacing a short, elementary argument by a reference to a deep theorem should count as a simplification.