Hagedorn wisdom debunked?

One of the papers that appeared today is by

• Dienes and Lennek.

They claim that the usual wisdom about the Hagedorn phase transition is incorrect. The usual wisdom is that the exponential growth of the density of states implies that at temperatures higher than the Hagedorn temperature, the second quantized spacetime partition sum diverges because the Boltzmann factor is not sufficient to suppress the exponential growth of the number of states. Alternatively, the Hagedorn transition can be seen as the appearance of a tachyon in the channel related by modular transformations.

Dienes and Lennek argue that this widely believed argument is wrong.

• In the UV picture, the divergence is wrong in string theory, they say, because it effectively leads to you to treat string theory as field theory with many fields, which means that the one-loop integrals over "tau" are made over "Im(tau)" being positive and "Re(tau)" smaller than one-half. The correct stringy integral involves the fundamental domain of the modular group "SL(2,Z)", and if this domain is used, the divergence at the (old) Hagedorn point disappears, they claim.
  
• In the IR picture based on the existence of tachyon, they note that the tachyon is actually projected out by the GSO projections, and consequently, it has no physical consequences. The usual Hagedorn phase transition is therefore absent in heterotic string theories and other tachyon-free superstring theories. They affirm that the Hagedorn behavior in type 0 theories, for example, still holds.
  

They also say that these two interpretations are consistent and related by a modular transformation. In another section of their paper, they propose that the Hagedorn transition in these non-tachyonic theories occurs after all, but at higher temperature than anticipated. The relevant temperature is not derived from a tachyon but rather from proto-gravitons and proto-gravitinos, as they call it. The resulting alternative temperature is universal for all string (heterotic and type II) theories, unlike the conventional Hagedorn temperature where the temperatures differ by factors such as "2" or "2-sqrt(2)". Their new universal Hagedorn temperature equals "a = 1/sqrt(2)" times the temperature that gives you the T-selfdual circumference of the thermal circle. You know that the conventional inverse Hagedorn temperature "beta_H" is not exactly the self-dual circumference under T-duality, and their new one is not either.

The authors also say that one of the conventional Hagedorn transitions does occur but it is a 10th order phase transition where "10" is indeed the spacetime dimension "D" (and it would be replaced by "D-1" if the dimension were odd).

I am curious what others think about this paper.