Below I will present an example of a finitely presented group which is conjecturally simple. Thank you for the opportunity to communicate this example to a wider community.

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Consider the group
$$\Gamma=\langle s,t,u \mid s^7=t^7=u^7=1, u=s^3t^3, u^3=st \rangle. $$
Then $\Gamma$ itself is not simple: it has a non-trivial homomorphism to $\mathbb{Z}/7\mathbb{Z}$, defined by $s\mapsto 1$, $t\mapsto -1$, $u\mapsto 0$.
However, the kernel $\Gamma_0$ of this homomorphism, is conjecturally simple.
Clearly, $\Gamma_0$ is finitely presented, as it is of finite index in $\Gamma$.
Much is known about the groups $\Gamma$ and $\Gamma_0$. For example: 

* These groups are infinite groups and they have Kazhdan's property (T).
* Every nontrivial normal subgroup in either group is of finite index.
* These groups are non-linear over any field.

These groups act cocompactly on an exotic $\tilde{A}_2$-building of thickness 3, so they are somehow related to the [Fano plane][1], the projective plane over the field with two elements, which hints about the role of the power 3 and 7 in the presentation of $\Gamma$.
The group $\Gamma$ is briefly discussed in section 10.4 [here][2].
It is closely related to the group 
$$\langle s,t,u \mid s^7=t^7=u^7=1, u=st, u^3=s^3t^3 \rangle $$
which happens to be an index 3 subgroup of an arithmetic lattice in $\mathrm{SL}_3(\mathbb{F}_2(\!(x)\!))$, thus residually finite.

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The above example is in fact a part of the infinite family of groups acting cocompactly on affine buildings. It is conjectured that if the corresponding building is not a Bruhat-Tits building then the acting group contains a simple subgroup of finite index.


[1]: https://en.wikipedia.org/wiki/Fano_plane
[2]: https://arxiv.org/abs/1608.06265