What is known about equiconsistency of PFA and existence of supercompact cardinals? Question: What is the last status of known partial results and new approaches on the problem of equiconsistency of PFA and existence of supercompact cardinals?
 A: We do not know that $\mathsf{PFA}$ and supercompactness are equiconsistent. In a sense, we are far from knowing, but in another sense, most set theorists who have worked on the problem are almost certain.
The problem is that the tools we have for deriving lower bounds in consistency strength are not strong enough to reach supercompactness (or even $\kappa^+$-strong compactness). As far as the current tools can reach, we know that $\mathsf{PFA}$ is at least as strong as that. 
Typically, lower bounds are established not by using the full power of $\mathsf{PFA}$ but instead by verifying that some combinatorial consequence of $\mathsf{PFA}$ already requires at least that lower bound. For years, the only combinatorial consequence of $\mathsf{PFA}$ that we could extract such strength from was the failure of square principles, first established by Todorcevic, in

Stevo Todorcevic. A note on the proper forcing axiom, in Axiomatic Set Theory (Boulder, Colorado, 1983), Contemporary Mathematics, vol. 31, American Mathematical Society, Providence, RI, 1984, 209-218.  MR0763902 (86f:03089).

Using this, Sargsyan proved in On the strength of PFA. I that if $\mathsf{PFA}$ holds, then there is an inner model containing all the reals and satisfying $\mathsf{AD}_{\mathbb R}+$"$\Theta$ is regular". This is stronger than the existence of a proper class of Woodin cardinals and a proper class of strong cardinals. Sargsyan's techniques already give us (a bit) more than this, and the bounds should increase as descriptive inner model theory reaches stronger determinacy models. This is the current best bound we have.
Recently, alternatives to the approach via square have also been established. Neeman (see also his work with Schimmerling) proved in

Itay Neeman. Hierarchies of forcing axioms II, J. of Symbolic Logic, vol. 73 (2008), pp. 522–542. MR2414463 (2009d:03123),

that the strength of $\mathsf{PFA}$ restricted to $\mathfrak c$-linked posets is that of a $\Sigma^2_1$-indescribable cardinal, and the restriction to $\mathfrak c^+$-linked posets should be what he calls a $\Sigma^2_1$-indescribable $1$-gap $[\kappa,\kappa^+]$. There is a natural hierarchy associated with the notion of $\Sigma^2_1$-indescribability, that ends up in supercompactness, so this gives us a new route to attempt establishing lower bounds. Already at the level described in the paper, though, the fine structural models required for the results are beyond what we can reach currently.
Another alternative was suggested in

Matteo Viale, and Christoph Weiß. On the consistency strength of the proper forcing axioms, Advances in Mathematics, vol. 228 (2011), no. 5, 2672-2687. MR2838054 (2012m:03131).

Their results show that the usual iteration techniques for forcing $\mathsf{PFA}$ require at least a strongly compact cardinal, and if the forcing used is proper, then a supercompact is indeed needed. This is not the same as saying that we indeed need supercompactness in strength, but it goes to explain the "almost certainty" I mentioned above. 

Two remarks need to be added, both related to the strength of failures of square.
I. 
There are many combinatorial variants of square principles, of varying strength, for instance the family of principles $\square_\kappa,\square_\kappa^2,\dots,\square_\kappa^{<\omega},\square_\kappa^\omega,\dots,\square_\kappa^\kappa=\square_\kappa^*$, and also $\square(\lambda)$ and its variants. I will not discuss here all known results; they are due to a number of authors, including (in rough historical order) Todorcevic, Magidor, Cummings, and Strullu. (The list is grossly incomplete, and I apologize for this.)
We have that $\mathsf{PFA}$ implies not just the negation of $\square_\kappa$ for $\kappa\ge\omega_1$ but, in fact, of $\square_\kappa^{\omega_1}$. It actually implies the failure of $\square(\lambda,\omega_1)$ for $\lambda\ge\omega_2$. The stronger principle $\mathsf{MM}$ reaches farther. For instance, $\mathsf{PFA}$ is consistent with $\square_{\kappa,\omega_2}$ holding for all $\kappa\ge\omega_2$, while $\mathsf{MM}$ implies $\square_\kappa^*$ fails whenever $\kappa$ has cofinality $\omega$, etc. 
A number of set theorists (notably, Magidor) have taken this as partial evidence that in fact strong compactness and supercompactness have different consistency strength, with strong compactness sufficing for the failures of square visible to $\mathsf{PFA}$, and supercompactness being responsible for the additional failures coming from $\mathsf{MM}$.
Together with $\mathsf{PFA}$ and $\mathsf{MM}$, in the last 30 years or so, a variety of reflection principles have been identified, some incompatible with $\mathsf{MA}$ (such as Todorcevic's Rado conjecture), some being a consequence of $\mathsf{PFA}$ (say, Moore's Mapping reflection principle), and some being consequences of $\mathsf{MM}$ (the Strong reflection principle, for example). 
In spite of their obvious differences and mutual incomparability, all these principles imply failures of square, and so they have significant consistency strength (the bounds discussed by Sargsyan). Unless all these principles turn out to be equiconsistent with supercompactness, a finer analysis than we can currently anticipate will be needed to separate their strength. 
The reason for the last remark is that the way strength is derived via failures of square can be understood as a proof by contradiction: If the principle under consideration does not imply the existence of inner models with certain large cardinals, then we can use this smallness assumption to build appropriate local core models $K$ (which are fine structural models and therefore satisfy square principles at suitable cardinals). We can then argue that the failure of square implies that these models must compute successors incorrectly. The point is that $\square_\kappa$ is upwards absolute between models where $\kappa$ and $\kappa^+$ remain cardinals. This is a violation of weak covering (one of the key properties of $K$), which implies that $K$ cannot actually exist, meaning that the large cardinals we were after were indeed reached during the attempted construction of $K$. 
An entirely different approach seems needed if (as Magidor suspects) these reflection principles end up having different consistency strength. In particular, weak versions of square are equivalent to the strongest known versions in fine structural models, so weak covering or its relatives cannot be enough to distinguish lower bounds between these principles. 
II.
Recently, we have begun to understand how single failures of square principles are significantly weaker than consecutive failures. In particular, the strength of $\lnot\square(\kappa)+\lnot\square_\kappa$ has been investigated. This started with work of Schimmerling and continued in 

Ronald Jensen, Ernest Schimmerling, Ralf Schindler, and John Steel. Stacking mice, J. Symbolic Logic, vol. 74 (2009), no. 1, 315–335. MR2499432 (2010d:03087). 

Independently of the study of square principles, the "stacking" technique described in this paper has been significant in modern investigations in inner model theory. The strength reached from the failure of both $\square_\kappa$ and $\square(\kappa)$ for $\kappa\ge\omega_3$ is comparable to that reached through Sargsyan's method -- the goals and techniques are different: Sargsyan looks at the failure of $\square_\kappa$ for a singular strong limit cardinal $\kappa$, and is directly developing the core model induction, while the techniques in the stacking paper are purely inner model theoretic and pursue showing the existence of non-domestic mice.
See here for Square principles in $\mathbb P_{max}$ extensions, work involving me, Larson, Sargsyan, Schindler, Steel, and Zeman. We look at the failure of $\square(\omega_2)$ and $\square_{\omega_2}$, in ($\mathsf{ZFC}$) forcing extensions of models of determinacy. Looking at strengthenings of this failure led to the isolation of the notion of $\Pi^2_1$ subcompactness as playing a key role. That this is indeed the appropriate large cardinal has been essentially confirmed in Equiconsistencies at subcompact cardinals, by Neeman and Steel. The goal here has not been to identify the strength of $\mathsf{PFA}$ or $\mathsf{MM}$, but rather of some of their fragments, below $\mathsf{MM}(\mathfrak c^+)$. We expect that a good understanding at this level will be key for an eventual understanding of the strength of $\mathsf{PFA}$ via an appropriate stratification (in this respect, the goal is similar to that pursued in the hierarchies paper by Neeman mentioned above).
